The Christmas Invasion - S2-E2
Corrected entry: A few times it is possible to see the Major's epaulettes. The symbol is a crown as in the British army. But seeing as Unit is a United Nations organisation (it stands for United Nations Intelligence Taskforce), an officer would have no reason to have a crown symbol on his epaulettes.
The Christmas Invasion - S2-E2
Corrected entry: When the Doctor wakes up, Hariet Jones and her translator don't have to read from the electric translation device any more because the TARDIS is translating for them. But they've never been anywhere in the TARDIS.
Correction: The translation ability is comprised of equal parts of the TARDIS translation circuit and the Doctor's natural Time Lord telepathy. The ability has been explained in the original series as something that a Time Lord can share with his friends. As the Doctor regained consciousness, since he was likely expecting to be surrounded by Rose, her mum, and Mickey (friends), he could have simply shared with everyone in the area.
The Christmas Invasion - S2-E2
Corrected entry: In the in-vision commentary, Julie Gardener's party hat keeps going from being on her head and off throughout the episode without time to move it.
Correction: It's hardly unreasonable that a commentary might not be filmed in one shot; there are many reasons why they might wish to stop or even go back from time to time. A degree of editing is to be expected and there is no pretence otherwise. As such, no "continuity errors" can be considered valid.
The Christmas Invasion - S2-E2
Corrected entry: The Sycorax begin speaking English for all of the humans because the TARDIS is suddenly activated when the Doctor wakes up. However, it is stated elsewhere in the series that the TARDIS only translates for those who have been in the TARDIS, which Harriet and her translator haven't.
Correction: The TARDIS translates for anyone the Doctor wants it to, including not only companions but people that he and his companions are dealing with.
The Christmas Invasion - S2-E2
Corrected entry: When the Sycorax leader is talking to the humans they need to use a translation device to understand him because the TARDIS isn't translating. When they talk to the Sycorax leader he seems to have no trouble understanding their language without any apparent translation device.
Correction: Just because it isn't "apparent" doesn't mean that it isn't there. After all, the TARDIS translation circuits aren't visible either. Safe to say that, as a star-faring species, the Sycorax would have some method of understanding the races that they encounter.
Correction: The United Nations has no military forces of its own, it seconds troops from member states. These troops wear UN insignia relevant to their mission (e.g. UNPROFOR in the former Yugoslavia) but their rank insignia is from whatever military organisation they belong to - in this case a British Army Major.