(Around 55 minutes) as Jackson says 'Bravo sir' the position of his left hand varies between by his side to holding the lamp post.
When the two Doctors are being pulled up the building on the rope, they are hanging by their hands; and therefore should be pulled through the open window head first. However, they are shown to fall through the window feet first, which is a physical impossibility.
The Next Doctor states that Cybermen are creatures from another world. In fact, Cybermen are converted humans. However, their origin in this series was on a parallel Earth, and their origin in Doctor Who (1963) was on the planet Mondos. Both of these qualify as "another world," so he is technically correct.
As the TARDIS (balloon) is ready to take off the assistant starts releasing the tether ropes. Despite being buoyant, the last tether tied to the post is still hanging loosely.
London is shown to be full of electric streetlights, invented in 1875. In 1851 they would have been gas powered.
The Doctor says he has never met his future self before, but many past special events of Doctor Who (1963) involved two or more time-displaced Doctors meeting each other.
(Around 24 minutes) when Miss Hartigan is testing the cybercontrol of the 4 men she tells them to turn right, which they do. She then tells them to turn left but they turn a full 180 degrees rather than turning back to face her. Obviously the actors knew the next line and acted accordingly but the order should have been 'Turn around' rather than 'Turn left.'