I am aware of that the polish discovered the machine in about 1933 and began their analyzing of it and eventually decrypted the code. I am also aware of that the British and French military met in Warsaw in 1939 and brought home 2 Enigma machines.
That being said, i read an article concerning the capturing of the submarine U-110 in 1941 during a North Atlantic convoy battle. In this submarine they should according the the author have found an Enigma machine. After further reading of several articles about this, some might say that the capturing of U-110 and from this, obtaining an enigma machine for the cryptologist's in Bletchley park, played the major role on decrypting the enigma machine.
So who decrypted the code - or did it even matter, obtaining an enigma machine from U-110, two years later?
There were numerous variations with Enigma (if you'll forgive the Elgar pun), both of the machine itself and the procedures for using it, over time and across the German Army, Navy, Air Force and other areas such as the police, railways and Abwehr (military intelligence). As a result there wasn't really a single person or team who cracked it, but a series of successes and setbacks; the initial successes of the Poles had been drastically curtailed by the introduction of additional rotors and changes in operating procedures in 1938/39, early work at Bletchley Park and Turing's bombes overcame many of those problems, but Naval Enigma proved more troublesome than Army or Air Force, in part due to the Navy introducing yet more rotors to select from.
In terms of captured materials it wasn't really Enigma machines themselves that were particularly useful, it was associated documentation; machine settings, procedures, related codebooks etc. The phrase "success breeds success" is particularly applicable to the work of Bletchley Park as bombes worked on the basis of a known-plaintext attack (a "crib" in Bletchley parlance), looking for a word or phrase that would appear in a message. Poor security discipline, such as an operator regularly transmitting a message saying "Nothing to report" at the same time each day, was a great help, and the more that could be gathered about networks, procedures and such, the more avenues there were to attack if something changed in the way Enigma was used. That's where U-110 comes in, though Hugh Sebag-Montefiore in Enigma suggests that, as heroic as the capture of U-110 was, the importance of the material recovered has been somewhat overstated; documents recovered shortly before from the trawlers Krebs and München were the real key to regular breaking of Naval Enigma, though further material from U-110 was very useful.
Breaking Enigma really was a team effort, as dull a conclusion as that may be.
Alan Turing is credited with breaking the Enigma Machine Code.