When The Doctor breaks into Magpie's he shatters the front door glass to get in. When he is shown leaving the glass door is completely intact.
When The Doctor goes to chase the undercover police car for the 2nd time on his scooter (when Grandma Connelly is taken) he is briefly seen with Rose's pink helmet in his hands, as if to put it on himself, but when next we see him (turning the corner into the fake 'dead end' street) he's wearing his own white helmet.
During the showdown at the transmitter tower, Tommy replaces the burnt out vacuum tube on the device and plugs it completely in. In the next shot, the plug is not completely in and Tommy's hand is seen plugging it in again.
When The Doctor first gets on his scooter, we see Rose putting her helmet on before she gets on, but she doesn't do her chin strap up. When we next see her, on the bike, her strap is done up.
When about to chase the black car, Doctor Who puts on Rose's helmet, pink with white visor, and in the next scene on the scooter he is wearing his own helmet, white with black visor.
This episode takes place on the day before and the day of Queen Elizabeth II's coronation (1 and 2 June 1953). However, at the start of the show (a scene set some time prior to the main events), the Connolly family are seen listening to the BBC radio comedy series "Take It From Here". The episode heard was first broadcast on 15 June 1953, two weeks after the Coronation.
When the Doctor and Magpie are climbing the transmitting tower, several insulators (brown and green structures) are shown attached to the tower with no wires on the other end. This exposed end should have the actual antenna wires attached. When Magpie connects the device, he connects it to the tower structure itself and not to an actual antenna. This would have grounded out the signal and done nothing.
One of the programs on the new television set is Animal, Vegetable, Mineral? (1952). In 1953 this quiz show was broadcast on Thursdays. The scene is set on Monday, 1 June, eve of the Coronation.
When The Doctor is climbing the transmitter, there is a shot in which he is viewed from above and is then struck by lightning. In this shot, The Doctor's foot disappears briefly as it leaves the green screen on the ground under David Tennant.
When the Doctor and Rose are chasing the black car, as they turn a corner just before they approach the closed gates, a modern lamp post can be seen.
When the Doctor is gathering stuff from Magpie's a G10 Glass-Epoxy circuit board is shown (green board on top of the pile). The paper based phenolic (brown) boards were used until the 1970s when they were replaced by the more fire retardant glass-epoxy boards.
A camera and a crew member are reflected in the Doctor's and Rose's glasses whilst they are riding on the scooter shortly after the opening credits.
The Doctor's psychic paper apparently tells the security guard that the Doctor is the king of Belgium, after which the guard lets him pass and asks: 'Shouldn't you be at the coronation, sir?' Kings are traditionally not present at the coronation of another king or queen regnant themselves. Instead, they usually send a representative (either their heir apparent or their spouse). Therefore, King Baudouin of the Belgians did not attend Queen Elizabeth II's coronation. He was represented by his brother, the future King Albert II.
Rose corrects Mr. Connolly who calls the British flag a "Union Jack" Rose says it's a Union Jack only when it's flown at sea and called the Union Flag otherwise. However in The Empty Child (2005), Rose referred to the Union Flag on her shirt as a "Union Jack" without correction.