At the end of the movie, Paul von Hartmann has tears on his face, with red eyes, when talking with Helen Winter. After a few cuts back and forth, his face is dry of tears, and his eyes are clearer. It's possible that the film-maker intended to suggest that some time had passed (for him to clean his face).
The SS officer Franz Sauer had rank insignia mismatched on the uniform. The collar tab had the insignia of a Hauptsturmführer (Captain), while the shoulder boards lacked the proper pips for the same rank, indicating an Untersturmführer (2nd Lieutenant)
When Chamberlain's plane takes off, the very distinctive outline of the Battersea Power Station is visible in the background, with its four chimneys. However, in 1939, when the film is set, only Battersea A was fully built, Battersea B being not yet complete, so the building at that time only had two chimneys.
The facade of the Führerbau in Munich shows very evident signs of wear and discoloration from decades of weather (actual state at the time of filming), although historically, the building had been completed only the year before (1937).
There were no gates at the start of Downing Street before Margaret Thatcher.
The liason aircraft used in the movie to transport Chamberlain is not period correct, probably to keep the budget under control. Its a Beech D18 (Twin Beech). It didn't exist in 1938/39.
At the Oxford ball, drunk Paul rolls up his pant legs and wades into the river, shouting "Behold the glassy waters of the Isis!".
He is not wading in the Isis (Thames) river, however, but almost certainly the Cherwell, a much smaller tributary of the great Isis/Thames and, unlike the Isis, does meander though Oxford Colleges.
When in Munich, Paul is clearly paranoid about not being heard or seen with Hugh, yet they both speak publicly, in German, about stopping Hitler's plans right next to a table of German military members. He even makes himself noticeable by borrowing their newspaper, yet no one addresses nor questions his actions.
When the assistant is face to face with Hitler the assistant draws his gun from his pouch, apparently intending to shoot the Fuhrer however he does not. In reality he would have been searched quite thoroughly by Hitler's security. No one except the very highest Nazi's close to Hitler would have been permitted to carry firearms into Hitler's company.
Paul and Legat meet in Munich at a Jazz club, where among the patrons are several Gestapo officers in uniform. While not technically illegal, Jazz music was severely panned by the Nazis as "degenerate music", and it would be very unlikely for a Gestapo officer to be seen in a jazz club, let alone in uniform.