~Bleach is property of Tite Kubo, Shueisha, TV Tokyo, and whoever else has the rights to it~
Ichigo encountering Rukia and being made a Substitute Shinigami jump started the entire series. So, how would things be different? This isn't going to be a narrative or story per se, but just an overview of how things would have played out from the beginning of the series to the current point in the Manga.
The two of them meeting seems like an unavoidable event, but let's just say through some cosmic coincidence Ichigo never sees Rukia, and she kills Fishbone D before it can even come near Ichigo's home. From there, Rukia would just continue as she would have, slaying any Hollow that entered Karakura Town with little difficulty. Grand Fisher is the only one that could pose any problem, but I'm almost positive she could have defeated him if she release Sode No Shirayuki.
After a while, she would return to Soul Society and another Shinigami would take over as Kurumadani did. Also, Byakuya would never have a change of heart and would likely never acknowledge her.
Now what about the other characters?
Urahara and Co would just go on as they had been, Kisuke continuing to hide the Hogyoku and remaining under the Soul Society's radar. Same goes for the Vizards.
Inoue would likely never awaken her powers as they were formed because of the Hogyoku(as far as we know). Chad would've awakened his powers because his are Fullbring, potentially being recruited by Xcution.
Uryu was always a Quincy and would've gone about his business. But that's a problem...his business is slaying Hollows. There's a high likelihood that Rukia would encounter him, so there are two things that could happen in that situation: (1)She could turn a blind eye, (2)or she could report him to the Gotei 13. In the second scenario, he would die. Soul Society would send some Shinigami and kill him, or worse capture him and be experimented on by Mayuri...and then die. Rukia would likely remain dutiful, and would go with option 2. So Uryu is dead...yay.
Now we get to Ichigo. This is tricky because he always had his own power(and a heck of a lot of it), so the question is...would he awaken it or not? Rukia in a sense activated it, as it was latent inside him until that point. However, as with Toshiro, we have seen Zanpakuto spirits call out to their owners even without them having an Asauchi. This may happen with Ichigo, causing him to train with Urahara or Isshin. Now, what Zanpakuto would he have? The obvious answer is his original Zanpakuto that he got in his shattered shaft training. However, with Isshin being involved, he would likely reveal the truth of Ichigo's heritage. Meaning that Ichigo might end up training to be a Quincy, which would cause harmony in his Soul and him gaining his current Zangetsu(dual wield). Yet even with these abilities, he would stay under the radar with the rest of them, training and learning Bankai and the such.
Now we go over to the Soul Society...where nothing is happening. Things we would be just as they were at the start of the Soul Society arc. Most of the Captains and Vice Captains wouldn't be training due to there being no war or major threat, and daily life would continue as it did for a 100 years. Aizen would be just lying in wait until the opportune moment. He was waiting for a very long time to defect, so he would just continue to be patient. As for the Arrancar? I'll talk about that next chapter.
A/N:
This story(sort of...not really...at all. Analysis?) was inspired by, well, me asking the question to myself. I found little to nothing discussing this, especially for what seems like such an obvious question in the Bleach fandom. So I decided to write this and share my thoughts on the scenario.
As I said, I will be covering everything from the beginning of the series to the current point in the Manga during the 1000 Year Blood War. So SPOILERS! Note that a lot of things during the final arc will be vague as it hasn't finished yet.