Since the release of the film The Imitation Game, it has been widely asserted that
The cracking of Germany’s Enigma code shortening the war by two to four years and saving an estimated 14 million to 21 million lives.
What is the source of this claim and how widely is this view held among historians?
The very breaking of Enigma - by Turing et al using Bombe and also by pinching of the German Naval codebooks - gave the British a blind spot that did nearly cost them the war. That blind spot was that German Naval Intelligence had broken the British Merchant Marine codes in 1938-9 and was reading transmissions using that code into 1944. The British never suspected, and never changed the code.
When times became particularly tough for the UK in Winter-Spring-Summer of 1942, their focus on fighting the U-boat through Enigma, either using Bombe or pinched codebooks, most likely distracted them from thoroughly investigating why the U-boats were so successful in locating convoys. Yes, the UK was in dire straits through mid-1942, but breaking the German codes was not the only way to fight that battle - changing their own codes would have been sufficient, and was an avenue still available even without Bletchley Park's achievements.
So in the final analysis, the achievements of Bletchley Park were only one of two clear paths to success in the U-Boat war; that war could still have been won without it - whether it would have cannot be said.
If the British Isles had been forced into surrender by starvation, it is hard to imagine that a D-Day landing, or defense of the Suez, could have been maintained. The Soviets could have possibly been forced into a peace by exhaustion if the Germans had another million or so men to defend the Eastern Front, as it is well known that they were running short of manpower by 1945.
As Andrew Hodges, the author of the authoritative Turing biography The Enigma reports, "the main serious source for this type of claim comes from a talk given at Cambridge in 1993 by Sir Harry Hinsley, the official historian of British Intelligence in WW2":
Now the question remains how much did it shorten the war, leaving aside the contribution made to the campaigns in the Far East on which the necessary work hasn't been done yet. My own conclusion is that it shortened the war by not less that two years and probably by four years - that is the war in the Atlantic, the Mediterranean and Europe.
His argument is worth reading in full, but is based on several lines of reasoning but begin with turning, in North African an "almost certain defeat into a stalemate", keeping Rommel "out of Egypt between his victory at the Battle of Gazala in 1942 and the British getting ready for their own victory at El Alamein" chiefly by "killing off his seaborne supplies" without which success, the Allies would have abandoned the operation against North West Africa, resulting wither in delays of at least a year of other crucial strategic efforts — notably the Normandy landings. Further these landings themselves could not have been conducted at the scale and with the success they were, without the successes at Bletchley Park, and to the extent that the Allies "wouldn't in fact have been able to do the Normandy Landings, even if [they] had left the Mediterranean aside, until at the earliest 1946, probably a bit later."
The premise of the question is incorrect, in that the Enigma was never "cracked". If you read "The Hut Six Story" by Gordon Welchman, you will find that Enigma messages could only be translated when operators made errors such as using the same key repeatedly or repeated use of the same base codes ("discriminants"). When the Enigma was used correctly it was unbreakable.
Nevertheless, many transmissions were intercepted and decrypted due to improper operational use of the machine.
The estimate of lives saved is based on the idea that the project shortened the war by "two to four" years, thus extrapolating that 2 years times 7 million deaths per year is 14 million. The origin of the "two to four" years idea I think is the 1974 book "The Ultra Secret" by Winterbotham.
While cryptography operations were certainly helpful, claiming that it shortened the war by two years is an overstatement. For example, in the Battle of Atlantic, the theater in which the decryptions were supposedly most important, I doubt anyone would characterize code breaking as the decisive technology. Radar, direction finding, convoy tactics, sonar, and anti-submarine aerial patrols, especially from Iceland, were all probably more important to the outcome than code breaking.
