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: When they show the soldiers killed by the Dalek, in both the hallway outside the cage and the large room in weapons testing, 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.
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'.
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)
Revealing mistake: When the alien spacecraft crashes into Big Ben, it is obvious that the shot has been reversed/flipped around. Look at the Roman numerals on Big Ben's clock face: They should read (clockwise from the top) XII, I, II, III, and so. Instead they read IIX, IX, X, etc. (00:06:35)
Revealing mistake: The 'earthquake' that takes place in this episode results in CGI-created cracks in the pavement that are very unconvincing. They do not appear to displace anything and - in a related continuity error - the cracks disappear in shots taken from ground level (the cracks are only visible in high-level 'looking down' shots). (00:34:30)
Plot hole: In "Aliens of London", it is established on a 'missing person' poster that Rose Tyler is approximately 19 years of age in March 2005 (The poster says 'missing since 6 March 2005'). This means that she must have been born prior to March 1986. However, in "Father's Day", which is dated 7 November 1987, Rose is still a baby in her mother's arms, when she should be a toddler aged approx two.
Revealing mistake: The Doctor and Rose locate the "metal" hatch that leads down into the Nestene Consciousness' lair. Watch carefully after the Doctor removes the "metal" hatch and moves it aside. The entire hatch pops up a second later. This is because it is plastic and it was suctioned to the floor.
Deliberate mistake: When Jack is scanning the gas mask people, he has a scanner on his right wrist. However, when he shows Rose and the Doctor his image of the Chula warship, the scanner is on his left wrist. The makers flipped the stretch of film, according to the commentary, to make the shot work. (00:37:45)
Continuity mistake: When the three bride Autons attack Jackie, and they open their hands one at a time, it is clear that when the second Auton opens its hand, all three are already open. After it cuts to Rose, and back to Jackie again, the third one opens (even though we just saw it already open). (00:38:15)
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. (00:16:30 - 00:17:08)
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. (00:40:33)
Revealing mistake: When Mickey closes the wheelie bin lid, he then finds the plastic stuck to his hands, he lifts his hands up and the plastic stretches like tar. If you look closely when he lifts his hands up for the first time, you'll see that a shadow of one of his hands is cast on his shirt, but there is no shadow of the strands of plastic stuck to his fingers. Additionally, when he turns around to pull away from the bin, he crosses the strands over each other, but they are still parallel in the following scene. (00:22:40 - 00:23:10)
Other mistake: At the end, the Doctor and Rose treat Charles Dickens by making the TARDIS dematerialise in front of him. The engines begin to grind, but the time rotor (the thing in the console's central column that goes up and down when the TARDIS is flying) does not move. (00:42:53)