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.
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.
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). The ship shown in the movie (the fictional 'RMS Archimedes) is a merchant ship so 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.
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.
All wet lead acid batteries release hydrogen when charging, this explosive gas gets trapped in the batteries and doesn't fully vent after charging, and can create an explosive localized atmosphere under many situations. The hydrogen is from the electrolytic action of the acid and the lead plates, having electricity run through them reforming the plates and releasing excess hydrogen from the H2SO4 (sulfuric acid) that develops as a result of the battery discharging.
Shortly after the heating system of the boat has stopped working, we can see the contents of a knocked over coffee cup being frozen solid. This could not have happened in anything remotely close to the movies time frame; the temperature of the seawater outside would have to be so low as to freeze the sea solid. Saltwater may be liquid at temperatures slightly below the freezing point of freshwater, but not as low as this.