Continuity mistake: In the scenes in Bletchley (south of England) it is obviously early Spring (daffodils, few leaves on trees), but towards the end of the film, and apparently only a few days later, when a German U boat surfaces off the coast of Scotland, it is clearly high summer, with all the trees covered in leaves. Trees don't come into leaf that quickly, and the seasons are always later in Scotland.
Plot hole: When the police search the white car and the barn they fail to find the compartment in the car behind the boot in which the decoding equipment was hidden. This is completely improbable. It was a standard car, the compartment was not a secret add on, and the police were supposed to undertake a thorough search of everything.
Other mistake: The last shots of Trafalgar square you can see twice a UPS van driving in the background.
Other mistake: In the 1946 Trafalgar Sq scene, modern double yellow lines can be clearly seen as being blacked out on the edge of the road.
Factual error: At the beginning of the film, as a British Admiral (or senior naval officer) leaves Bletchley he returns a salute. Although a naval officer, he does not give a naval salute but rather a half hearted army salute.
Continuity mistake: In the scene in Scotland where Puck is rendezvousing with the German submarine whilst being watched by the British security services and police, there are several shots of the sub through binoculars supposedly from the perspective of the British on top of a hill. These views are all (with the exception of the last one of Puck meeting the sub) from sea level.
Factual error: The scene of 1946 London - Trafalgar Sq. looking south down Whitehall. You can see the turrets of the new MP office building - put up only a few years ago.
Factual error: In the opening scenes the staff car door is opened by a four-ringer Wren (Deputy Director, very rare) equivalent to Royal Navy Captain, instead of a rating.
Character mistake: According to Alan Turing, Enigma has "150 million million million" starting positions (1.5 x 10^20), while Shark has "4 thousand million billion", or 4 x 10^18). So Shark, the "ultimate refinement" of Enigma, has FEWER starting positions? (00:10:25)