Factual error: The Doctor (and the staff on the station) refer to the idea of a planet being in orbit around a black hole as 'impossible'. It is not. Gravity (and physics in general) works perfectly well outside of the event horizon. For all practical purposes - regarding orbiting around it - the black hole might as well be any other object, as long as it has an equal mass. (00:07:15 - 00:08:30)
Other mistake: When the Doctor tells the Atraxi "Oh, you're not the first lot to have come here, there have been so many," one of the monsters shown from their scan of Earth is a group of Ood. Even if in one of their uprisings they attacked the Earth, it would have been in the far future. The Doctor has had no encounters with them on Earth, and he knows them to be peaceful, so why would they be amongst the aliens who have threatened the planet? The only Ood who has been to Earth at this point is Ood Sigma, and he didn't come to attack. In addition, a group of Hath is also shown. These aliens have never been seen on Earth at all. (00:53:30)
Continuity mistake: As the scarecrow-soldiers advance on the school to do battle, the scene cuts to Sister of Mine, looking out from a school window. There is heavy rainfall outside. When we immediately cut back to the advancing scarecrows, it is a cold, clear night. (00:10:00)
Factual error: One of the programs on the new television set is Animal, Vegetable, Mineral? In 1953, this quiz show was broadcast on Thursdays. But, it's the eve of the coronation, which makes it Monday the 1st of June. Animal, Vegetable, Mineral? wasn't on that evening.
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.
Continuity mistake: Miss Kizlet wears a large silver brooch on her suit's right lapel in every shot of her, but after Clara notices the lights going out behind her house, when it cuts back to the office Kizlet's brooch switches to her left lapel just before it cuts back to the Doctor and Clara, and since the brooch has a distinctive design we can tell that it's a flipped shot. (00:25:11)
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'.
Continuity mistake: When the Doctor and Ida descend into the core of Krop Tor, the screen charting their progress shows 39% Oxygen. After cutting away to an overhead shot of the two, the camera cuts back to the same descent display, only now the oxygen percentage (which has been descending constantly all this time) is at 42%. (00:35:25)