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.
Deliberate mistake: The final Auton attack in this episode is supposed to take place in a busy London street, but it is a pedestrianised area, and several shots reveal this. For example, there are no road markings visible, no apparent distinction between pavement and road, and a bus stop appears to sit in the middle of the 'road'.
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.