In the early scenes of Dunkirk evacuation, a Union Jack is shown flying on a British ship. Ships of the Royal Navy flew the white ensign and merchant ships the red ensign.
Spitfires, models or replicas perhaps, appear in German markings.
In the Dunkirk scenes in the beginning of the film, most of the rifles carried by the British troops are not Lee Enfield 303's, judging by the sling fitting they might well be German rifles or possibly Italian ones.
Nobody, military or civilian, carries a gas mask.
The diagonal stripes on the Union flag flown on the ship in the Dunkirk scenes do not meet at the intersection of the vertical and horizontal stripes as they should do but are several inches too high on the top half and and too low in the bottom half. The design of the Nazi flags hanging on the walls also have errors.
The "Spitfires" in RAF roundels are actually Hispano Buchons, a Spanish-built derivative of the Messerschmitt 109, while the Messerschmitt 109s are portrayed by Spitfires (either real or models) painted in Luftwaffe colours.
Londoners hurrying to an air raid shelter in 1940 pass a wall sign for the Royal Festival Hall, opened in 1951.
In the opening scene, the British are shown using a bazooka-like weapon to knock out the tank column. In 1940, there was no such thing as the weapon seen here.
When the conspirators are on the terrace of the pub, the Telecom Tower can clearly be seen in the distance to the left of Tower Bridge. Construction on the tower didn't begin until 1961.
In a pub scene crates can be seen clearly marked 'Watney Mann', Watneys did not combine with Mann until 1958.
A London bus wouldn't show LONDON on the roll sign for the route's endpoint, but rather a specific place within London.
A loudspeaker message in the RAF control room reports "Five squadrons taking off from Calais heading North North East. Possibly heading towards London". London is North West of Calais, and NNE would be heading towards Paris.
At one point the lead character is seen on the south bank of the Thames, walking East, past the Houses of Parliament, towards Westminster Bridge. The scene then cuts to him still on the south bank of the Thames with the Tower Bridge in the back ground and with the cannons that sit outside the Tower of London clearly visible. It is an impossible correlation as the Houses of Parliament are 2.7 miles to the West of Tower Bridge. To be correct he would have to walk past Tower Bridge before getting to the Houses of Parliament. He must therefore have been walking backwards.
Sgt. Mulligan gives the order "On the double!" This is an Americanism; the equivalent in the British Army would be "AT the double!"