The Royal Navy captured and used two German U-boats. The first, U-570, was attacked and forced to surface by British aircraft on her first war patrol, on the 27th August 1941. Her inexperienced crew surrendered to the circling aircraft, and she was captured by a small flotilla of RN vessels, including the trawler Northern Chief and destroyers Burwell and HMCS Niagara. She was towed northwards to Iceland, where she was beached (she had been taking on water due to damage inflicted by the aircraft, and the beaching was necessary to prevent her sinking). On the 31st August, a British salvage team boarded her, repairing her lighting system, and shoring up holes in her hull. Once this was complete, she was towed to the destroyer tender HMS Hecla at Hvalfjörður, where more thorough repairs were carried out. At this base, she was thoroughly searched for any intelligence materials, and appreciations made of her capabilities by both the RN and USN.
While the repairs on Iceland were ongoing, the RN was at a loss as to what should be done with her. While there were suggestions that she be handed over to the Americans, or given a crew from one of the Allied navies, she would ultimately be taken into British service as HMS Graph (the fact that the name began with G signified a German origin). At the end of September, she would be sailed to the UK by a British crew. In the UK, she was completely repaired. She would then undergo a series of meticulous trials, to fully determine her capabilities, and any changes needed to Allied ASW tactics. The RN was impressed by her engines, hydrophones and diving abilities, but were scathing about her cramped living conditions, radio equipment and periscope. Once the trials were complete, the RN put her onto active service. She made her first patrol from Holy Loch on the 8th October 1942. On the 21st, she would engage the U-boat U-333 off the northern coast of Spain. While four torpedoes were fired, all missed. Graph did not pursue, as the crew mistook the sounds produced by damage previously done to U-333 for the breaking-up noises made by a sinking ship. She returned to the UK on the 29th October, setting out on a second patrol on the 19th November. This would again take her to the Bay of Biscay, where she would be ordered to intercept an Italian blockade runner, the Cortellazzo. Ultimately, Graph would not locate her target, which would be sunk by the destroyer Redoubt. Her final patrol began on Christmas Eve 1942. She was to patrol the Norwegian coast, in support of Convoy JW-51 to Murmansk. The only targets she would encounter were two German ASW trawlers, which she engaged, but missed. On her return to the UK, she would assigned to training duties. The RN was finding it difficult to keep her in service due to a lack of spare parts. Additionally, her engines and batteries were in dire need of replacement, as the Germans accepted shorter service lives in return for better performance. In February 1944, she was removed from service completely. Following use in tests of depth-charges, she was towed to the Clyde for scrapping. En route, she would run aground off the coast of Scotland after her tow rope broke in a storm. In 1947, she would be partially salvaged, but the remnants of her hull are still present, and can be visited by divers.
Graph was not significantly more advanced than her British or American equivalents. While there were some areas in which German practice was ahead, there were just as many areas in which Allied practice had the lead. To find an example where the RN used a German submarine that was significantly more advanced, we have to look at the other U-boat that saw significant service in the RN. In June 1945, following the German surrender, the British salvaged the Type XVIIB submarine U-1407. She had been fitted with an air-independent propulsion system running on high-test peroxide (HTP), which allowed her to make high speeds underwater. Once she was raised, she was taken to the UK, where she was fitted with a new set of machinery by Vickers, under the guidance of the original German technical team. Her torpedo tubes were removed, as she was not intended to be used as a warship. Instead, she was to be used to test the HTP system, and as a fast ASW target. Once the refit was complete, she was taken into service as HMS Meteorite. Throughout 1946, she carried out several trials, which convinced the RN that the HTP concept was worth further investigation. Two further HTP boats were constructed, named Explorer and Excalibur. However, the concept would never enter active service - HTP was far too volatile (There were several attempts to use HTP in torpedoes, which would lead to the loss of HMS Sidon, and more recently the Russian submarine Kursk). Meteorite was officially deemed to be 75% safe, while Explorer and Excalibur would be nicknamed Exploder and Excruciator by their crews. Meteorite would be scrapped in 1949, once she had outlived her usefulness to the RN.
Sources:
C.B. 4318(R) Report on "U-570" (H.M.S. Graph), Intelligence Division Naval Staff, Admiralty, 1943
Report on the German Submarine of the U-570 Class Captured by the British in August 1941, Office of Naval Intelligence, Internal Publication, 1941
The Führer Led But We Overtook Him, Phil Durham, Pentland, 1996
HMS Graph (P715), uboat.net, http://uboat.net/allies/warships/ship/7751.html
Nelson to Vanguard: Warship Design and Development 1923-1945, David K. Brown, Seaforth, 2012
Cold War Submarines: The Design and Construction of U.S. and Soviet Submarines, 1945-2001: U.S. and Soviet Design and Construction, Norman J. Polmar, Brassey's, 2005