Continuity mistake: Towards the end of the movie, when Alan comes out to Joan, there are a number of shots over his right shoulder. On the back of his jacket is a piece of lint that appears and disappears from shot to shot.
Plot hole: There is no logical reason to switch off the Bombe at midnight when the codes change. Firstly, Turing's proving the concept of automated code breaking, so even if it only finds the right settings days or weeks later, the experiment is worth doing, then you can work out how to speed it up to be operationally useful. Secondly, even in steady state operation, cracking "yesterday's" settings (and thus intercepted signals) is still going to be pretty useful in most cases. Threatening to smash the Bombe up at midnight is good stuff to add some movie tension but, in reality, it's nonsense.
Factual error: The opening graphic establishes the year as 1951 when depicting the time Turing's home was robbed and he was arrested for indecency. The robbery and arrest actually occurred in 1952, and strangely enough the correct date is listed on the dispatch about the robbery handed to MI6 boss Menzies moments later. Even if one looks at the film as a work of fiction, the date on the on-screen prop is inconsistent with the on-screen graphic. (00:00:40 - 00:02:40)Vader47000
Factual error: The voice-over incorrectly describes the Post Box as a 'Trash or Garbage can', when an envelope is placed in the box, and later retrieved by someone using a key, towards the end of the film.
Factual error: The railway carriages shown in a wartime scene are post-war British Railways Mark 1 carriages. Inside the train a compartment is labelled 'Standard Class' which did not exist at the time.