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: 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.
Continuity mistake: When the Doctor is looking at Rose's hands, turning them over repeatedly, while she's describing how she was hanging from a barrage balloon, there's one moment where her hands are turned palm-up in a shot facing her, but back-up in a shot facing him. (00:18:25)