In the U.K., this was released on D-Day, June 6, 1944.
This started life as an Army training and instructional movie, The New Lot (1943), written by Peter Ustinov and Eric Ambler, and which contained many of the cast members of this film (David Niven came in later). The training movie had upset some Army brass with its frankness, and was suppressed. It has recently re-emerged thanks to a copy found in an archive.
At the time the movie was made, David Niven, who played a Lieutenant, was a British Army Major serving on operations in World War II. He was a 1930 graduate of Sandhurst, the British Army military academy.
Peter Ustinov, who wrote the film and played the Anglophobic café owner Rispoli, was only a private in the British Army, so to explain his presence on-set to visiting senior officers, he was said to be Major David Niven's batman (officer's servant, a kind of military valet). Ustinov served with the British Army Film Unit for most of the war.