Revealing mistake: Whenever you glimpse the inside of the TARDIS from outside, a frontal view reveals a picture of what's inside which has been placed at the back of the box. Sadly, it isn't on the sides. This means that, from the right angle, you can easily tell that it is just a box.
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)
Continuity mistake: When the Doctor pulls the arm off the Auton and tosses it to Rose, the arm is bent at the elbow and solid plastic, with a smooth, clean break. Later, particularly noticeable when Rose has left the department store, the arm is hollow, with an unbent elbow and a jagged break. The arm continues to change between the two appearances for the rest of its screentime. (00:07:08)
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)
Revealing mistake: When the Doctor fires a champagne cork into Auton-Mickey's head, the lack of spraying or leaking wine, even when the Doctor is seen holding the open bottle horizontally immediately afterwards, makes it quite clear that the champagne bottle is actually empty. (00:25:23)
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)
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'.
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.
Revealing mistake: There are some moments during the scene where the Doctor is being strangled by the Auton arm where it's fairly obvious that Christopher Eccleston is holding the arm to his own throat. Namely, the moments where the hand isn't quite in the right position, slipping towards his chin.
Continuity mistake: After the Doctor freezes the Auton arm, its index and middle fingers are curled, while its ring and pinky fingers are straightened. When the Doctor and Rose are in the stairwell, most noticeable when they emerge at the bottom, the fingers of the arm are now all slightly curled.