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.
Continuity mistake: When Bonnie throws the laptop with the Osgoods' message to the floor and stomps on it, the computer's position changes between shots - take note of the tape markings on the floor. In addition, the position of the plastic frame of the laptop's screen changes: in the first shot, it's mostly popped out, but in the second shot, it's mostly attached.
Continuity mistake: For most of the robbery sequence at the beginning, the Nightmare (a disguised Ashildr) is holding her pistol in her right hand. When she's yelling at the Doctor for interfering with her robbery while he's looking at the chest strapped to the back of the carriage, however, she's holding it in her left hand. It switches back to her right after the carriage escapes.
Deliberate mistake: In real life, Viking helmets didn't have horns on them, because it would be a liability in combat. The horns were actually for drinking. The Vikings in this episode were given horns anyway, because the writers didn't feel it was worth having to include an explanation of hornless helmets to a general audience.
Continuity mistake: During the early scene with the Cybermen outside St Paul's, when Missy pulls out her phone/disintegrator weapon and shows the Doctor pictures of the Cybermen going viral all over the planet, and when she prepares to take a picture of the two of them, her grip on the device, as well as which hand she's holding it in, keeps changing between shots.
Continuity mistake: When Courtney comes into the caretaker's room in search of paper towels, there is a "keep out" sign (reading "Go Away Humans") hanging in the window of the inner door. When the Doctor is talking to her just before she leaves, the sign is missing. Over the course of the rest of the episode, the sign proceeds to appear and disappear a few more times.