(Until I can get to a source text, or a better-qualified answer turns up:)
The Enigma coding system was used at some level by all branches of the Reich's armed forces. Each section used a slightly different variant and different codes of the day: Army, Air Force, surface Navy, U-Boats.
It had the advantage of being usable in almost any situation: the machine itself was portable and battery-powered, and encryption/decryption could be done at a pinch by a single operator, or by two operators (naturally it was associated with the radio ops).
Of these, the U-boat service used the most secure procedures for generating codes-of-the-day: they used rotors from a larger set, and the plug board settings were changed in a more secure way than the other services. Turing was mainly concerned with the U-boat system, and the different services' systems were dealt with by different teams at Bletchley, organised into different 'Huts'. The movie ignores all but the Hut 8 team, and shifts personnel that it wants to include in the drama into that Hut.
The operation at Bletchley was vast, involving thousands of people, and there were eventually hundreds of Bombe machines. The Bombe was a development of a machine developed by the Polish cryptanalysts before their country fell, but their method failed when the Germans improved their procedures just prior to the War. The use of guessed plaintext passages was crucial to the design of the Bombe from the start and was not the moment of breakthrough depicted in the movie. Turing supplied valuable insight and new designs which allowed the British Bombe to work despite the German improvements. The first new Bombe was actually operating in early 1940, before Churchill became prime minister. Further improvements by the Germans caused the system at Bletchley to falter badly, and there was a devastating period in 1940/41 when no U-boat information was being decrypted. But information extracted at considerable cost of life to the British, from a captured U-boat, was successful in getting the whole operation back.
The U-boat Enigma decrypts did allow the British to know all the orders followed by the U-boats as to which routes to take, which areas to patrol etc, whenever the Bletchley system was working. There are diagrams in Hodges' book which show a time line of the Bletchley decrypts against Allied Atlantic shipping losses, on a month-by-month basis, which show directly how important Enigma decryption was. The supply convoys across the Atlantic were vital to Britain's survival.
The movie ignores a second code, known as the Lorentz cipher, or by code name Tunny to the British, which was a far less portable machine which encrypted or decrypted teletype messages directly. It was used only for communications between general headquarters but carried very high-level communications, often top-level orders from the highest levels of the Reich to coordinating Generals.
This was also decrypted at Bletchley, by a separate team, although Turing gave helpful insight into the underlying mathematics as well. This was the code that allowed those privy to the secrecy of the Bletchley operation to feel like they had a spy in Hitler's private office, and was invaluable in the later War, especially at D-Day.
Would you be able to clarify your question please? It can be interpreted in two different ways:
Did the enigma machine passed to the allies by the Polish have as large an influence on WWII as the movie shows
Did the machine that the allies built to decode Enigma - [bombe] (http://en.wikipedia.org/wiki/Bombe) and the information gained using that machine really have as large an influence on WWII as the movie shows?
I suspect you are asking about 2 but would like to make sure.
You may be interested in the answers to the previous question, "The Imitation Game" ends with the quote "Historians estimate that solving the enigma code shortened WWII by 2 years". Which historians claim this? Why and what is the evidence for it?. The response don't try to definitively answer your question, but at least gives you some more context.