It's worth nothing I've taken some liberties with the time lines. This is the Tenth Doctor, sometime in series 2, probably before the Cybermen stuff. Regarding Call of Duty, I've gone on the assumption that Price and Soap returned to Britain after the events of Modern Warfare, and that this takes place in the five years between then and Modern Warfare 2.

"About half a kilometre to the contact point, Captain."
"Roger that," came the reply, "We've no intel on codename TL, we don't know what he looks like, what he's offering, where he's from – hell, we're only assuming he's male. Keep a sharp lookout and follow my lead."
"Aye, Captain."
The two men began their ascent up the next cliff face, the captain leading the way. By the time they reached the top, they were visibly exhausted and getting out of breath, but the captain, a much older man sporting a thick moustache, lit a cigar and took a deep pull.
"Filthy habit, Soap, I should really give it up. Not as young as I used to be."
"Ah, you only live once, sir," Soap replied, "In our line of work, you gotta take every little pleasure you can."
"That's the truth," the captain agreed. He took another pull on his cigar, and looked thoughtful, "Half a kilometre from here is nothing but open tundra. There's no city for thousands of miles – the nearest village is two hundred and seventy miles. Who would arrange a meeting out here?"
"I guess that's what we're going to find out, sir."
"And on that note, let's move, shall we?" the captain said, "Progress won't be quick in this blizzard. Follow me."
The two men made the distance in good time, but it wasn't until they were less than fifty metres from the meeting point that visibility allowed them to see their contact.
"Sergeant, can you see it too?" asked the puzzled captain.
"Aye, sir." came the reply. Soap was looking down the scope of his rifle at the strange thing, "Looks like a blue box, wooden. What's a 'Police Box?'"
"What's it doing out here? And why's it in English?" The captain was intensely alert. "Move up, keep an eye out for movement and fix your sights on that box. I don't like this at all."
The box was about a metre square at the base, and roughly 3 metres tall, with what looked like a blue light on the top. It did indeed seem to be made of wood, and each side had two windows in the panels, at about head-height. It stuck out like a sore thumb against the bleak, white landscape, at the top of this mountain. The captain ordered the other man to look inside, but he couldn't open the door.
"Seems to be locked, Price. What the hell's going on?" he exclaimed.
Captain Price paused. "I don't know, Soap. There's nothing else out here and it's not like we've got any other leads." He looked at the box. "Have you tried knocking?"
"What?"
"Knock on the door."
"Right," Soap muttered, not apparently convinced that this was a clever idea. Tentatively, he rapped on the door of the Police Box and stepped back. There was a rattling from behind the door, and the two soldiers raised their rifles at it and stood, motionless. It opened and a tall, thin man stepped out, wearing nothing more than a suit and a long coat. It was twenty degrees below. He looked at the two men in puzzlement, then grinned widely.
"Hello, you must be the UNIT boys, pleased to meet you. Do come inside, it's rather chilly out here!" he said, approaching the soldiers.
"Stay back!" Price barked, "Identify yourself!"
The man paused.
"Sorry, yes, of course, I'm the Doctor. Could you put the guns away, I really don't like guns?"
Soap and Price shared a glance.
"I don't think so," said the captain, "Where's TL? Give the detection phrase!"
"We're warning you," Soap added, "Identify yourself or we will fire."
"Yes," said the man, apparently unconcerned, "The silent owl flies past the windmill. Now can you please put those guns away? Sorry, what were your names again?"
Price lowered his rifle, but his sergeant kept his aimed at the strange man.
"But the bats stay hidden in the barn," he replied, "Who the hell are you and why did Colonel Mace send us a thousand miles into the middle of nowhere to meet you?"
"I quite want to know what he was doing in that box," said Soap.
"Oh, a Scottish lad!" the tall man said with a large smile, "Don't meet many of them really. Love the Scots. Strange food though. Lots of frying."
"Look, TL," growled Price impatiently, "I've had just about enough of this. Either you tell us why a UNIT colonel had us sent out here to deal with your bloody games or I'll shoot you right now. Talk!"
"No, I told you, it's the Doctor. That's my name!"
"Doctor what?" the two soldiers asked in unison.
"Just the Doctor. Do you mind if I bring my associate out here? Only she'd love to meet you. Only...can you please put the guns down, I don't want to ask again."
"Lower your weapon, sergeant," ordered Price, "I don't think we're gonna get anywhere with this guy by threatening him."
The man stuck his head inside the door and called "Rose!" He shouted as though the person he was beckoning was a distance away, as if in a different room, though the other person couldn't have been more than a couple of feet inside the box. After a few seconds, a second person stepped out of the box, wearing clothes more appropriate for the climate – furs, hats, scarves. It was hard to tell underneath all of the clothing, but they seemed to be a girl, probably early twenties, if not younger. When she spoke, this was confirmed.
"Doctor, are we gonna be here long?" she said, her voice muffled by the wind and scarves, "Only you promised you'd take me to see the Beatles." She looked at the soldiers, who were visibly puzzled by the "Doctor's" associate. "Gotta love a squaddy," she said with disdain.
"Who are you two? How the bloody hell did you get out here dressed like that?" demanded Soap, his patience wearing thin.
"More to the point, what's so special about your information that we had to trek all the way out to the middle of Siberia to get it?" Price asked aggressively. He put his hand on the grip of his rifle and walked towards the couple from the box.
"I'd better get some fucking answers soon."
The tall man looked serious.
"Right, yes, of course," he said, "What I have to tell you is vitally important for the future of the entire planet. This must reach -"
There was a wooden, splintering thud, followed by the sound of a gunshot. The girl, Rose, screamed and the tall man ducked.
"Sniper!" Soap barked, both of the soldiers holding their guns to the ready, scanning what little they could see.
"You two, get down now!" Price ordered to the two civilians. They leapt into the snow at their feet, but the man was apparently more concerned with with the wooden box than the danger they were all in.
"They shot my TARDIS! With bullets! It's got a hole in it now! I don't even know if it can repair the outside any more!"
"Shut the hell up!" Price hissed, "Stay down and stay quiet!"
There was zip of a bullet flying past, and another gunshot.
"Muzzle flash, seven hundred metres, ten o' clock," whispered Soap, as he aimed his rifle in the direction.
"Get in the TARDIS!" said the "Doctor."
"What?"
"The blue box, get in the box!" cried the girl, and the both of them stood up and ran into the wooden box.
"Oh, you've got to be shitting me," Soap said in disbelief. He took a shot at the sniper. "Movement, sixty yards left!"
Price got on one knee and fired at two indistinct shapes in the snow, his carbine loud but muffled in the blizzard. One of them dropped to the ground, and the other crouched, clearly taking aim. Price shot the other one, and they also dropped.
"We're in the shitstorm now, Soap. Baseplate, we're taking fire and we need evac now!"
"Copy that Captain," came a voice over the radio, "Baseplate enrou – shit, I'm seeing multiple heat signatures in the snow, look like SAM sites. We're gonna have to bug out, we're dead if we stay here."
"What? Oh, I don't believe it!" the captain said with dismay.
"Sorry, Captain. Nothing we can do. Good luck."
"Good luck."
"Soap, we're on our own," he said, keeping his eyes fixed on the surrounding area, "We're gonna have to go to plan B."
Soap grunted his acknowledgement, then, "What about those two?"
"Those two can go to hell, we've got nothing out of them. Our main priority is to exfil the area safely."
"Come on you two!" came a voice from behind them, "Are you going to stay out here all day? Get inside now!"
"Are you insane?" hissed Price.
"Get in! Trust me, it's safer than out here," he said, beckoning them into the box.
The soldiers looked at each other.
"Looks like we don't have much choice." Soap admitted. Price followed the man into the box, and Soap, taking one last look around, went in too.
"What the bloody hell..."

I have a vague idea where I want to take this, but don't expect regular updates or anything. If anyone is interested, in my mind, Price is armed with an L119A1 UGL, and Soap is armed with an HK417