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: When the TARDIS crash-lands at the beginning, it comes to a halt in a corner of the Powell Estate's courtyard, by the garbage containers. Later, when Rose is watching through the window as blood-controlled people walk through the courtyard, the TARDIS has moved further out into the plaza. (00:23:05)
Continuity mistake: At the beginning of Rose and Jackie's conversation in the kitchen about the Doctor, Rose takes a piece of food out of the fridge. Moments later, when she rests her hand on the light switch by the door, it has vanished. At no point in the intervening time did she come anywhere near a spot where she could have put it down, and she didn't eat it either. (00:04:10)
Continuity mistake: When Daniel Llewellyn meets Harriet Jones, she pulls out her ID card when she introduces herself. The scene cuts to a shot of Llewellyn saying that he knows who she is, and when it cuts back to Harriet, the ID card has vanished without her making any movement to put it away. (00:16:10)
Continuity mistake: After the Sycorax ship has arrived above London, when Rose has decided that they need to shelter in the TARDIS and is explaining this while preparing to put a housecoat/dressing gown/bathrobe on the unconscious Doctor, the garment jumps between the bed and her hands in a few shots. (00:30:25)
Factual error: The episode kicks off with a reuse of the zoom-in from orbit first seen in "Rose." As the sunny North Pole indicates, the Earth is depicted as if it is the height of Northern Hemisphere summer, even though as the title of the episode indicates, it begins on Christmas Eve, three days after the northern Winter Solstice when the Pole is in darkness. (00:00:05)
Continuity mistake: When the robotic Christmas tree attacks Rose, Jackie and Mickey in the Tylers' flat, it barrels through the internal window and destroys a supporting pillar separating the living room from the main hallway. Later in the episode, the pillar is seen intact. (00:13:30)
Other mistake: When Rose and Mickey are running across the plaza after being dropped off by a cab, a fence is visible between where they are and where the cab pulled up. It is unlikely to say the least that they would have gotten out of their cab right by a fence and climbed over it. (00:08:45)