The British ship wouldn't have flown the Union Flag (often called the Union Jack) at the stern as this is never used as an ensign (the flag at the back of the ship). Assuming the ship shown in the movie was a merchant ship then it would have flown the Red Ensign which is red with a small Union Flag in the top corner nearest the flag pole.
US fleet subs of that era were incredibly cramped and would not have contained as much leg and head room as the one depicted in the movie. The submarine set used for interior shots is built on a larger scale to accommodate both actors and the shooting crew.
When they come out of the sub from the forward torpedo room they walk up a set of stairs. In submarines during the war there were no stairs out of a ship they climbed up ladders through hatches. What they walked through was really the forward torpedo loading hatch that was opened and stairs added to the sub after it was turned into a museum ship to make it convenient for tourists to enter and exit the sub.
Flooded batteries release hydrogen because as they reach full state of charge, some of the energy goes to split the water into hydrogen and oxygen instead of causing the chemical changes that charging involves. A discharging battery does not release hydrogen. Therefore, the only time there would be a hydrogen buildup is when they were on the surface charging the batteries. They would have other problems - like CO2 build-up - but hydrogen building up to explosive levels? No.
The use of grappling hooks was not an anti-submarine tactic used by the German Kriegsmarine during World War II. That tactic was used by German and British naval forces during World War I, but only in shallow water. There are also reports of the Japanese Imperial Navy using hooks in the Pacific during World War II.
The diving officer orders the boat to periscope depth, then adds, "66 feet." On a WW2 fleet boat, minimum periscope depth was 55 feet, optimum was 58 feet, and maximum was 62 feet. At 66 feet the scope would have been 4 feet under the surface.