Factual error: The rescue team at the MI6 safehouse takes ages to arrive. A spec-ops team with helicopters should have been on alert and arrived in minutes.
Deliberate mistake: Where are the motorcades for the other world leaders?
Plot hole: They find the terrorists by the electricity they're using, but they're using generators, off the grid.
Other mistake: There has been a major terror attack on London. Explosions big enough to send shockwaves across the capitol, and yet people can be seen walking about as if nothing has happened. (00:29:00)
Character mistake: When the Vice President has worked out Banning's signal he says "All right Manning" instead of Banning. (00:48:50)
Suggested correction: I hear him say "Banning."
Factual error: Air Force One has landed. The caption reads: "Stansted airport, 20 miles north of London." it's closer to seventy miles. They probably wanted to get to the city quicker. (00:17:20)
Suggested correction: It's 38 miles to drive down the M11 from Stansted to St Paul's cathedral - and about 20 as the crow flies.