When Churchill visits an RAF base, and the squadron is scrambled, a Spitfire and a Hurricane are shown taking off as a pair. In fact, Hurricane and Spitfire operations were kept separate, and the two types would not have been mixed at (and scrambled from) the same dispersal, as depicted in the film.
When Churchill is dictating his famous "Never Surrender" speech and walks from one room to another, a modern door closer can be seen on the top center of the frame.
Clement Attlee was played by a Scottish actor with a Scottish Accent. In fact he was born in Putney, South West London and therefore did not have a Scottish accent.