Factual error: Churchill tells his aide to fetch "Air Vice-Marshal Dowding". Dowding was actually an Air Chief Marshal, two ranks higher, and wears the correct rank insignia for that rank when he is seen later. As a keen military man, this is not an error Churchill is likely to have made.
Factual error: As Pug and Byron are readying themselves for dinner at the White House, they are discussing the Bismarck. Pug refers to the ship as a "pocket battleship", which it most definitely was not. As a navy captain, Pug should have known the difference between a pocket battleship like the Scheer or Graf Spee, and the real thing.
Factual error: The Royal Navy lieutenant aboard the American warship is wearing a moustache. Royal Navy personnel must be either clean-shaven or wear a "full set" (beard and moustache). They are not (and never were) permitted to wear a moustache alone.