Okay, for my staggeringly high amount of readers, not to mention my loyal flock of followers (/sarcasm_off), this is what's going on:
I've got everything up to Chapter 5 lying on my HDD, and it's been that way for the past month or so.
Trouble is, I was almost shocked to death by how egregious my writing happened to be in those chapters when I looked at them last week, so I'm starting to re-write it all as of now, trying to right my wrongs and evade possible public embarrassment (Jan 4th).
The story won't change at its core, not that you've seen a lot of it at this point, so don't worry, this is still the tale that I started that long while ago, just told differently. And without further ado, here's my reworked first Chapter. Anyone willing to critique this would be greatly appreciated for their effort.

The Doctor did not remember it all, but one thing was very clear throughout.

All of it happened way too quickly.

One second, the Time Lord was standing in his TARDIS, setting the zigzag plotter into its neutral position whilst calculating the optimal approach vector to his target,—61 Coram Street, Bloomsbury, London, England, on the 18th December of the year 1897 AD—the next, he was flung backwards, hit his head on an unknown object, was blinded by some inexplicably bright light, fainted and only woke up afterwards, and lying helplessly on the ground at that, to the sight of 61 Coram Street, Bloomsbury, London, England, on the 18th December of the year 2017 AD.

"Well, that's just marvelous, is it not?" the Doctor asked no-one in particular.

Did I just say 'is it not?' How eloquent have I become? Too much for my own good, that's how.

As his last regeneration was still a recent memory, the Time Lord's own self still felt uncomfortably new and undiscovered, like someone utterly foreign to him.
Although he'd previously decided on using his memories of past incarnations for guidance, he couldn't help but feel a slight sense of discomfort when doing so, as if he was plagiarizing other people's work and accomplishments, and this was only made clearer now, as their alleged wisdom had thrust him into a minor disaster.

Surveying his environment as all the noise and hubbub had subsided, the Doctor was quick to realized a number of things that were quite obviously, inexplicably wrong. There were the unusually modern-looking automobiles droning along in the distance, for one. Then, there was the awfully conveniently-placed street sign to the Doctor's right, towering above him. Its appearance fit into the supposed late 19th-century ambiente, but it clearly read 'Maple Street', not 'Coram'.
The Doctor, being a Time Lord with significant experience and carrying great wisdom, was quick to reason that he had made a navigational error. So far, so explicable, one might say. There was just one tiny problem, one thing that arguably changed the importance of this scene tremendously, that transformed it from an unlucky mishap to a major catastrophe.

His TARDIS was gone.
Of course, it had seemed strange from the getgo to him, to arrive not inside of his trusty vessel, but lying on the floor, unconscious, and with only limited recollection of any events past, but the Doctor, despite being as intelligent and experienced as he was, also had a fresh face to get acquainted with, and it had quickly chalked the paranormality of all this up to a separate, as of yet undiscovered event, certainly not related to his journey to England, which was but to visit a friend of his. Perhaps the regeneration had even corrupted his memory, he thought. Wouldn't be the first time, aftet all.

Making sure that the TARDIS really was nowhere to be seen and not just closeby, waiting for him somewhere around a corner, he turned around multiple times, surveyed the entire area, fiddled with his sonic screwdriver, snapped his fingers and even called the time-traveling starship by its name, but to no avail. Putting two and two together, the Time Lord surmised that he must have been the unfortunate victim of some spacefaring alien entity's attempt at kidnapping one of the universe's most famous, not to say infamous time and space travellers. Yet again.
And so, with a quick rub of his cane—it fortunately had neither disappeared nor taken any damage, but exhibited a strangely high amount of dust—he strode on forwards, not spending too much time thinking about directions or planning.

If this is anything like the last time I…a Doctor lost his TARDIS, then it will just show up at some point. No need to worry about it.

Just stay calm, enjoy 21st century Earth for a bit, and let the clues come to you, not the other way around.

His positive outlook on things definitely helped with the search, though it's hard to say by how much. So, as he thought it, he did try to enjoy the scenery and stay calm, with rather mixed success.

Walking through London's streets for the umpteenth time, recognizing many, if not all of the sights on his way, the millennia-old Time Lord began to feel something that he, much to his despair, recognized from memory. It was a something he had neither hoped nor expected to experience.

Every time one of London's inhabitants passed him, the feeling went up, going from the depths of his stomach all the way to the throat, throbbing with the rhythm of his twin hearts beating.

The Doctor had already had some experience with the feeling, even knew its name, but forced himself not to think about it, hoping it would go away on its own. He remembered how other incarnations of himself had coped with it, even if some of them had to go through a lot of hardships to get there. It comforted him, knowing that more than a dozen versions of himself, most of which he would consider emotionally or physically weaker than him by far, in some cases both at the same time even, had succeeded in defeating the inner foe that he was battling then and there.
And so, the Doctor strode on.

He continued eyeing nearly everything around him; from buildings to animals, vehicles, people, even plants. After all, the malevolent entity who surely set this up could be anything, anything at all, right? That's the problem with having a memory that goes back thousands of years, he thought to himself: You cease to be able to recognize large patterns, because at this point, you've seen it all. Everything is just as random and out of the blue as everything else.

Any other Doctor would have now at least begun thinking about getting impatient, if only slightly, but not this one. He simply continued advancing through London's streets, keeping his pace, direction and, most importantly, his attitude, constant and fixed.
Even though he did collect a few strange looks this way, it paid off.

"Well, what do we have here?" the Doctor asked as he finally stopped for once.
Precisely, he was referring to the apparently infinitely long and many hundreds of meters high thing in front of him. 'Thing' is the term of choice here, as any effort made describing it would be quite a waste. A peculiar sight to say the least. He went back a few seconds in memory, only to discover that he had no explanation for how he couldn't have seen such an enormous structure any sooner. So, he went a few steps backwards, physically this time, accidentally bumping into an English citizen, an old man of about sixty who wildly gestured and yelled at him in response. The Doctor made no efforts to calm the man down, knowing how it would be a futile effort anyways. He waited for him to walk off in anger and the nearby pedestrians, currently giving the Time Lord quite uncomfortable stares, to get back to minding their own business, which all happened within a minute or two. He got back to his original plan and intent; the thing from before. He looked up. Nothing. Nowhere was anything paranormal to be seen.
Business as usual. The Doctor slowly made his right leg rise—thankfully, he had preserved his heading from before, so he knew he was now going straight towards where the building had been moments earlier—and put it down on the ground before him, then his left. It took some time, and he trembled slightly all the way, even if he'd probably never admit such a thing, but it was eventually accomplished. And lo and behold, the same structure appeared once more.
Very clever use of a perception filter, he commented, as if the makers of the anomalous building were there to take his critique.

With the same arduous, careful, and, truth be told, scared pace as before, the Time Lord took another step forward, his cane clacking on the brick floor as he did so. Reacting nearly immediately, the scenery took on even wilder forms. What was previously just a barely describable mess of grey and brown shades, like an already featureless object viewed through a thick, fogged-up glass lens, now appeared as a line of distinct, very Earthern-looking buildings, which were situated right in the middle of a sort of piazza, which in turn was not just completely unknown to the Doctor, not just completely physically detached from the rest of the city—as in, it was hovering in the air a few centimeters apart from the nearest solid object, despite easily fitting in judging by its Victorian-esque styling—but also, via a quite sinister-looking, rusty street sign, labeled as 'Bad Wolf Square'.

That certainly is not supposed to be there, he thought.
And I like it a lot.

His curiosity set on fire and mood skyrocketing, the Doctor swiftly jumped over the small gap separating the regular city floor from Bad Wolf Square's—both looked identical at first glance, so there couldn't be any danger, could there?—and hurried towards the ominous structure.

There, he was strangely disappointed to find out that it was not, as he had assumed, any kind of illusion, hologram or trickery. Nevertheless, he went over the facades with his cane and screwdriver a few more times, just to make sure. Then also came what he called the "solidity and structural integrity test", which involved but knocking on all visible surfaces, at least the ones he was able to reach, that is—which brought to his attention the bottommost row of buildings, which were rather unaesthetically cut in half, their windows and roofs meeting his waistline and continuing almost seamlessly onto the next row above, which in turn melted into the one above it and so forth, but anything below a certain point in height was simply nonexistent, it seemed, ridding the Doctor of examining the lower portions, let alone entering them through any door that might have been reachable this way—in order to make sure they weren't gaseous or permeable.
Just in case. Turns out they were not.

In fact, sooner rather than later, the Doctor came to the unfortunate conclusion that what he was looking at and knocking on really was an impossibly long row of impossibly high buildings right in the middle of 21st century London.

Analyzing them closer from the outside didn't help much. The architectural style was reminiscent of most of the normal buildings nearby and didn't follow any particular pattern.
From office blocks to semi-modern flats and even what seemed to be a hotel, everything was there, just randomized and overly tall. And stretching outwards to infinity to the left, right and above.

Why here…why drop me here without my TARDIS, only to place this, whatever the hell this is, right in my path… Maybe that's just it. This was deliberate. There will be some clues lying around somewhere in this place. There must be. I just have to find them.

And so, our brave wannabe detective began his quest to find out what was behind the great mystery of Bad Wolf Square. To do that, he needed to analyze the Square itself more closely, moving away from its main attractions.

There is a reason they called it Bad Wolf Square, not Bad Wolf Street or Bad Wolf Avenue.

The space besidesbuildings must be important. They didn't just waste it.
Didn't put it there for no reason.

In accordance with his plan, the Time Lord took a good look around, ignoring the buildings for the time being and instead focusing on everything else contained within Bad Wolf Square. There was, of course, the street sign displaying that name. He walked up to it, only then noticing that it was but single-sided, which meant it had been purposefully rotated in order to face him from the direction he had confronted the Square.

Which meant that its creator had known the Doctor's exact position at at least one point.

I'm most definitely dealing with some advanced form of intelligence here. Not just a space anomaly or a coincidence or some random, poorly thought-out Dalek plot.

Further investigating, the Doctor noticed the peculiar attributes of Bad Wolf Square's floor.
Although he'd called it average and uninteresting, in fact just fitting in with the remaining scenery just fine earlier on, as he know kneeled down, investigated it up close, he re-evaluated his previous conclusions, rejecting nearly all of them.
Whilst the section of London he had traversed earlier in order to arrive at the mysterious Square was, characteristic of most 21st century Earthern large cities, relatively uninteresting in this particular department, offering not much more than tarmac, rubber and concrete in their respective shades of boring grey, though there was some noteworthy old-age cobble and brick to be seen, the floor now supporting the Time Lord's shoes was made out of brick and stone, and varyingly exotic kinds of them at that.

This one…comes from Tivoli. As for that one…well, that's definitely native Earth rock, but not from the right time. It's at least nineteen-thousand years younger than its supposed to be.

None of this makes any sense. The clues—if you can call them that—they are random.
No correlation between each other. There must be some reason for that, some underlying idea or plan that ties all of this together in a meaningful manner. But I am not finding it.
And it's making me go mad!

Ironically, just after that train of thought had ended, just as he rose up in frustration and almost resigned from his quest completely, something caught the Doctor's eye, something, he hoped, that would help him solve this mystery after all.

"Hey, take a look at this," he cheered, almost, as he made his way towards the strange object lying on the floor a few yards away from him.

And stopped right afterwards.

It was not because of what he was seeing—he was still too far away from the object to make out any real details besides its small size and glittering, reflective appearance in the sunlight—but because of what he was hearing. Specifically, himself.

Who am I talking to?

And this is the moment where that feeling the Doctor had so vehemently tried to suppress made itself noticeable again. Not just that, a sheer eruption occurred within the Time Lord's mind, a tsunami of thoughts. Emotions of this level were not something this Doctor had any experience with, nor did he particularly want to. But on the other hand, there was no alternative.
He simply opened up to his inner desires and let it all out.

Or, for a lack of a better term, the Doctor was getting very close to losing it.

No, this is not loneliness! This is pure solitude. Solitude. I don't need anyone or anything to make me feel better or any such petty thing at all!Never have. Those…other doctors just had special needs. They needed their silly 'companions' for some sort of sense of gratification—but I am above them. I am past them. I am in every single way superior. I am the next generation!
The only reason I feel this way is because I'm missing my TARDIS. I'm homesick, that's what it is.

I shall find it quickly, and when I do, I will go on with my life pretending this monologue never happened in the first place.

Of course, that didn't happen. The poor Time Lord spent a long time going back and forth and back again deep within his mind, playing out entire debates and making long, dreary attempts at motivating himself for what felt like many days, but fortunately in reality accounted for but a few dozen minutes.

And in doing so, he forgot entirely about that peculiar thing on the ground he took notice of earlier.