When The Doctor takes to the Nightclub stage to ask if anything has fallen from the sky, he has his right hand and arm up and gesticulating. But in several other shots of him from behind, it is his LEFT arm that is up and moving.
When Rose looks around Jack's ship, right after her "Very...Spock" line the flats of her hands are seen. Here they are completely unharmed. Later, they are burnt.
Captain Jack is first seen at a social gathering of various military officers. It is night time, with an air raid in progress, yet the windows are not blacked out. Due to blackout regulations during the war, it was a criminal offence to show a light after dark (even if there wasn't an air raid).
Barrage balloons used during the Blitz were tethered with steel cables, not ropes.
When Captain Jack first introduces himself to Rose, he states he is a Captain in the RAF.
The RAF has the rank of Group Captain, often referred to as Captain, and has 4 stripes on the epaulettes. This is equivalent to Colonel (British Army) and Captain (Royal Navy). However, Harkness is wearing the rank of Squadron Leader (two thick, one thin stipe), two ranks lower than Group Captain, and equivalent to Major (Army) and Lieutenant-Commander (RN).]
The "psychic paper" was introduced as to be showing what the user (i.e., the Doctor) wanted it to show to his mark. Now suddenly it shows the marks whatever they want to know. He never said it only had one function.
When Rose and Jack are on the top of his ship the time on Big Ben is 9:30. The time never changes, despite the scene lasting several minutes.
When Jack is scanning the gas mask people, he has a scanner on his right wrist. However, when he shows Rose and The Doctor his image of the Chula warship, the scanner is on his left wrist. The makers flipped the stretch of film, according to the commentary, to make the shot work.
A plump family are seen taking cover in an Anderson shelter, but the shelter has merely been dumped in the garden, and not installed. It should have been sunk several feet into the ground and covered with earth, enabling it to withstand flying shrapnel.
There were no tape recorders - a major plot device on-screen - in the UK during the Blitz. Ironically, these had been introduced in Germany only at this time.
Nancy seems to have the unique ability to whistle with her fingers in her mouth - even though she's wearing gloves.
In the opening scene, when the two officers are looking out of the window, the army officer is wearing his '37 pattern web belt upside down with the buckles at the bottom.