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No. It was never intended to be a documentary. It's an action/suspense film based on submarine warfare during WWII. Inspired by actual events, as the majority of WWII movies are. The actual U-571 was never captured by either the British or the Americans - she was sunk off the Irish coast by a torpedo launched from a Shorts Sunderland Flying boat from No. 461 Squadron, Royal Australian Air Force.
(It should be noted that U-570 was captured by the British on her first patrol and saw service as HMS Graph from 1942 until 1944, when she was withdrawn from service due to maintenance issues).
The Enigma machine was actually first captured by the British, a fact the filmmakers acknowledged in the end credits of this movie. Edit
Out of the thousands of hours logged by U-boats, there is only one recorded incident in which a German captain gave the order to fire on torpedoed survivors. U-boat crews were more humanitarian and often provided aid to their victims before submerging. They very rarely brought survivors with them as space on a boat was extremely cramped for its own crew; often times the U-boats just simply left them to die, as seen in Das Boot. So the shooting of survivors in this film could be seen more as mercy killings rather than leaving them to die.
In terms of the film, it's likely that because U-571 was so badly crippled, short on crew members and supplies, with sensitive information aboard, it was too risky to leave the British survivors to possibly report their position. Not to mention, as we see with Taylor and his crew keeping Wassner alive; sabotage is a possibility as well. Edit