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.
Factual error: In the opening shot of this episode, we see the Earth from space. The camera then zooms into Western Europe, the UK, London, then into Rose Tyler's flat, where a jump cut to an alarm clock shows it's 7:30am. But look again at the start of that sequence: it shows that it's daylight over the UK - and over the USA. This is of course impossible. When it is 7:30am in London, it is 2:30am in New York, as Eastern Time is 5 hours behind UK time. In New York at 2:30am it is not daylight, because it is the middle of the night. In addition, later episodes make it clear this episode is set in March. The North Pole is fully sunlit, which it never is in March, due to the equinox that happens that month.
00:00:50 - 00:02:00
Factual error: When Rose is locked in the room and the bodies come toward her, you can see a modern-day electric light-switch to the side of the door she's trying to get through. When the Doctor hears her screaming for help and runs for her, you can see a central heating radiator. In fact, the undertaker's house has at least two radiators in different hallways. Both are wildly anachronistic for the time period.
Factual error: Emergency Service vehicles in Cardiff (indeed, throughout Wales) have their signs in bilingual English/Welsh. So the police cars and ambulances seen in this episode should have 'Heddlu'/'Police' and 'Ambwlance'/'Ambulance' bilingual logos. However, the police car seen after the earthquake has its label in Welsh only.
Factual error: Look closely at some of the freight wagons at 'Limehouse Green' railway station: several of them appear to have 'British Rail Freightliner' livery. This episode is set in 1941, but British Rail did not exist until 1948, and the Freightliner Division did not exist until the 1980s.