So for no reason I can see, the Baron tries to requisition the Doctor's TARDIS key, and they do a lot of You Old Bastard finger-wagging in each other's faces, and Churchill's all, "Think of what I could achieve with your remarkable machine, Doctor! The lives that could be saved!" Which, was Churchill known for coveting remarkable machines and I'm not getting it? Is there a reason he would be sneaky and creepy about the TARDIS? I am fully willing to admit it; my historical knowledge is entirely limited to things like murdered heiresses and who Lord Byron fucked and the details of most battles of Alexander's campaign against Persia, but other than that it's just a whooshing chasm filled with pop music and celebrity meltdowns. Even still, I never got the sense that Winston was the type to steal your time machine.
"So you've changed your face again," Churchill says, lest we get confused about basic shit on how this show operates, and the Doctor goes, "Yeah, well, had a bit of work done," which makes no sense either in his context or Winston's, but whatever, and then they (Amy) are so happy to be in the war room they can't stand it, and Winston bitches because he called the Doctor a month ago but he's just now showing up. (This timey-wimey problem of the TARDIS doesn't seem to extend past this episode, if you were wondering; I got really anxious thinking it would happen all season or else I wouldn't mention it.) But why did he call the Doctor? We'll never really know.
Churchill's staff of high-ranking military officials is mostly women and disabled minorities, as was the custom in those days, but that doesn't stop him from getting all up in one lady's grill to ask her what's wrong -- nothing -- and to yell Harkonnenishly in her face, "Action this day, Breen! Action this day!" She smiles at Amy nervously before leaving, again for reasons we may or may not figure out later, and takes off. A dude approaches with a report or whatever, and Churchill finally explains why the Doctor's here. Sort of. Except without the "explaining" part.
"I have something to show you. We stand at a crossroads, Doctor! Quite alone, with our backs to the wall. Invasion is expected daily. So I will grasp with both hands anything that will give us an advantage over the Nazi menace." I mean, I guess we're drawing a connection between the TARDIS and the Ironsides -- yes? -- and just how desperate things got there, for awhile, but again: We do this by making Churchill seem gullible and any-means-necessary, like, if in real life he'd had a TARDIS or a Dalek he would have used it. Which -- again, taking into account my lack of historical knowledge -- seems a bridge too far, considering neither of those things exist and anyway, it's us Americans that proved we can't be trusted with our toys.