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: In the scenes at Kings Cross railway station in London, overhead wires are visible above the train; Kings Cross did not have these until 1975.
Factual error: At the end of the film, the statement that the enigma code breaking was a secret for "over 50 years" is incorrect. The secret was only kept until the 1970s.
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.
Factual error: There is no evidence that Turing was aware that Cairncross was passing Enigma decrypts to the Soviet Union.
Character mistake: In the movie it is stated that "There are 159 million, million, million possible Enigma settings. All we had to do was try each one. But if we had 10 men checking one setting a minute for 24 hours every day and seven days every week, how many days do you think it would take to check each of the settings? Well, it's not days; it's years. It's 20 million years." This is massively understating it - if it would take 10 men only 20 million years to check all the settings of Enigma they would have to each check 251,861 combinations per second. (00:15:04 - 00:16:46)
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 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)
Factual error: At the one of the movie they say "Turing's machine" inspired generations of scientists. "Today we call them computers" suggesting he invented the computer, which is wrong. Despite there being different opinions. The first computer was built around the beginning of the 20th century. It's very far fetched to claim he inspired the modern computer, if not bogus. (01:50:10)
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.