Continuity mistake: While the Doctor, et al. are in the pyramid with the Monks, when the Monk tells the Secretary General that if his consent isn't pure it will kill him Xiaolian is standing right behind him, which can be seen in shots facing the Secretary General and shots facing the Monk, but in the one shot of the Secretary General as he's being vaporized Xiaolian has vanished yet Xiaolian is still standing in the same spot in the shot facing the Monk. (00:26:10)
Factual error: When the Doctor regrows his hand during the swordfight, he says that he can do this because he's still within the first 15 hours of his regeneration cycle, and there's still residual energy in his body. However: In both this scene, and the episode's opening scene, when the TARDIS arrives in London, it is full daylight. The episode takes place on Christmas Eve and Christmas Day, 3-4 days after the Winter Solstice. London, at 51°N latitude, has a night longer than 15 hours at that time of year, meaning the Doctor should have been out of luck.
Continuity mistake: While Clara shouts, "I don't know where I am," over and over, the Doctor sonics his way into the house. When he kneels beside Clara, then notices the 'girl' on the stairs and stands up, in the shots facing him we can see that the young actress stands on the step facing the stairway (note her legs), with her hand at her side, but in the shots facing the girl she stands facing the door with her hand on the banister. (00:15:22)
Continuity mistake: After the Doctor yells at Handles for putting him on the Dalek ship, the Doctor grabs him and carries the head with his right arm, then switches the head to his left arm. However, in the close-up as the Doctor asks, "What does this question mean?" we see Handles is back on the Doctor's right arm, but next shot as the phone rings, Handles is carried on the Doctor's left arm again. (00:01:56)
Other mistake: The security guards shoot away the lock on the door, which they then kick down with no real effort. But the door would still be held in place by the hinges - they don't kick it with enough force to rip the hinges out, and even if they had, we can see that there's no damage on the hinge side of the door anyway to demonstrate their forcible removal. (00:24:20)
Continuity mistake: When Lawrence is watching the Weeping Angel, just after seeing the Doctor's film/Easter egg, he turns away for a split second. By the time he turns back, you see the Weeping Angel's face only inches from Lawrence's face. But you then cut to a wide shot where you see the Angel's arms outstretched, so its face is about half a metre away from Lawrence's face. (00:34:45)
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)
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 two Cybermen meet the Dalek for the first time, they are exterminated. They fall directly in front of the Dalek, but when the camera angle changes to show the second Cyberman hitting the floor side on, you can see the Dalek in the background. The Cybermen both fell straight forward, so it shouldn't have been possible to see the Dalek when seeing the Cyberman side on in the next shot, as the Dalek is facing them both directly.
Continuity mistake: After Miss Evangelista is killed, we see her communicator as her "ghost" slowly fades. In one shot we see from a slight distance the lights left lit tick down to 3, then a short while passes, we cut back and the lights are back to 4. They then tick down to 3 again.