Factual error: Magnetic tape recorders weren't available outside of Germany during World War II. True the tape recorder could have been captured from the Germans, but, as it was "cutting edge" technology for 1941, it would hardly be likely to have ended up in a hospital office.
Revealing mistake: Dalek is escaping and enters the large room with the soldiers. He starts the sprinklers so he can kill them faster. When they show the dead soldiers they show some casings which were ejected from the soldiers' guns. The casings still have projectiles in them. If used, they would have been in magazines and would never fall out like they are shown.
Doctor: So, that's the trap. Or the test or the final judgment, I don't know. But if I kill you, I kill her. Except that implies, in this big grand scheme of Gods and Devils, that she's just a victim. But I've seen a lot of this universe. I've seen fake gods and bad gods and demi-gods and would-be gods - out of all that - out of that whole pantheon - if I believe in one thing... Just one thing... I believe in her.
Doctor: Pfft... I'd have to settle down. In a house or something, a proper house with... With... With... With doors and things. Carpets! Me! Living in a house! Now that... That is terrifying.
Rose: You'd have to get a mortgage.
Rose: Oh yes.
Doctor: I am dying. That's it. I am dying, it is all over.