When the Military Police search Turing's office at Bletchley Park, the same clip of a policeman emptying out a desk drawer is shown twice.
At 39:55, Hugh throws a glass at Alan's computer, leaving debris on its surface; however, just seconds before (at 39:48) when Alan defends the computer, the debris is already present.
After Alan is appointed leader of the group at Bletchley Park, he draws up a crossword puzzle to be published in the newspaper to recruit potential new team members. He pins this puzzle to a wall to show the remaining members, but the grid pattern and clues of the actual test crossword that was published in the Daily Telegraph on January 13, 1942, are clearly different from the puzzle that Alan pins to the wall. The actual puzzle from the newspaper is the one shown being worked on by several hopefuls in the movie.
When young Alan is called into the Headmaster's office to be told news of his close classmate, it is after the holiday break. Yet the headmaster tells Alan that the math instructor had caught Alan and his friend passing notes "a few days ago." The note passing could not have happened a few days prior when the boys had been gone on holiday for at least a few weeks.
At around fourteen minutes, the first messages that Turing's team are supposed to work on are generated by a (later) Enigma machine with four rotors. The machine of the team has only three rotors.
Alan Turing didn't design the machine by himself. W. Gordon Welchman, a mathematician not mentioned in the film, collaborated with him.
Naming the first bombe machine "Christopher" is a fabrication of the film. The first bombe was actually named "Victory".
Joan was not hired after solving a crossword puzzle in the newspaper. She was at Bletchley Park already when she was promoted to work with Alan Turing's group. Turing, however, did publish a crossword puzzle in the 13 January 1942 London Daily Telegraph in an effort to recruit more code-breakers.
Bletchley Park was much larger than depicted in the film. At its peak, over 9,000 people worked there.
The Polish intelligence did not obtain the Enigma machine from the Germans, but in fact Polish mathematicians led by Marian Rejewski managed to break an early Enigma code in 1932, and gave the British their own reconstructed copy of the Enigma machine and a "bomba" cryptological machine in 1939, which gave the British a starting point.
Joan and Alan are seen lying on the grass, discussing mathematical operations which would become the basis for the RSA algorithm for public key cryptography in 1977. But this is not an anachronistic "goof" by the filmmakers, but a deliberate Easter Egg - something for viewers "in the know" to figure out - tying the activities at Bletchley Park in WWII to the almost unbreakable codes which protect everybody's details in the Internet age. The mathematics they are discussing definitely pre-dated the setting of the scene.
Joan Clarke describes "Euler's theorem" but pronounces it incorrectly. She pronounces it "YOU-ler" not the correct "OIL-er". This seems unlikely as she has a "double-first" degree in Mathematics. However, in the 1940s pernickety correct pronunciation of foreign names was not a noted characteristic of the English, and it is also likely that the Germanic "Oiler" pronunciation was simply not as widely known at that time, even amongst knowledgable mathematicians - especially since Germany was the enemy. (Euler was Swiss.)
While the group is trying to decipher the daily Enigma code, the wall clock strikes midnight setting off an alarm. In frustration, Hugh throws all of the days work to the floor because the German military change the Enigma code daily at midnight rendering the work useless. Although Berlin is 1 hour ahead of London, in Secrets of Her Majesty's Secret Service (2014), Bletchley Park historian Dr Joel Greenberg states that midnight for the code change was Greenwich Mean Time.
The newspaper article that describes Turing's conviction for indecency misspells the word trial as "trail." However, a typographical error in a newspaper article was far from uncommon.
During the victory celebration, the crowd is shown cheering and waving a British flag ("Union Jack"). However, the flag is upside down, an international distress signal. Of course, when people are waving a large flag around, they are liable to do it any which way. The rule for which way up the flag is relates only to when it is flying from a flagpole.
The police inspector had a stubble beard. In those days you were clean shaven any facial hair was well trimmed. Beards were full. Stubble wasn't a thing until recently.
Not including Gordon Welchman into this movie is a major character error. He was a vital person in decoding the enigma machine. Without him, breaking the enigma codes would have been to late and history would be that the Germans won the war.
A few examples of Americanism in speech which would not be used at that time. Turing twice uses the adjective "smarter" which only meant 'dress neater', British people would use "cleverer" until decades later.
When Alan Turing is introduced to Hugh Alexander, it is said that Hugh Alexander is a national chess champion - twice. Although he really won twice the British national chess competition (the first time in 1938), the second time was only in 1958, many years after the end of the Second World War.
Detective Nock says that two professors became "radicalized" at Cambridge before they joined the communist party. But the term "radicalized" was not used in this way until the 1960s, when it was used in reference to civil rights groups.
In 1951, Nock uses typewriter correction fluid to change the name on a warrant to that of Turing. The first correction fluid (Liquid Paper) was invented in 1951 and was not available commercially until several years later.
Close to the beginning of the film the word "hiring" is used, when Alan is taking his induction. There is no way that this word would have been used at that time in that context. The more appropriate word at that time would have been "recruiting". The word "fired" is used when Alan and Commander Denniston argue about the Christopher's usefulness, midway through the film, the word used circa 1940 would have been "dismissed". The current phenomenon of using (Amercian) English, as opposed to (English) English in the UK is only a modern day trend. In the 1940s and 50s hiring and firing was not in the English vocabulary, at least in regards to employment.
In the radio message copying scene, the Morse characters heard are OJJB, but the copier writes KJJB on her pad.
When the code breakers break their first message on the Bombe, Hugh Alexander reads out the German submarine position as 53 degrees 24 minutes North, 1 degree West. This position is actually right in the middle of England, just east of Rugby.
Both the burglary at Alan Turing's home and his subsequent investigation for homosexual activity (which would have been referred to legally as "gross indecency") are shown investigated by Manchester Police. Turing's home was not in Manchester but in Wilmslow, Cheshire, and so both offenses would have been investigated by Cheshire Police (or 'Constabulary'). That his subsequent conviction was at Cheshire Quarter Sessions confirms this.
Alan Turing's Manchester computer (Baby, not another Christopher) was constructed at the university, not in his home.
The ship in which Peter Hilton's brother supposedly served was the HMS Carlisle. The HMS Carlisle was a C-class cruiser of the Royal Navy, spending much of the war in the Mediterranean prior to being damaged beyond repair in 1943. It was not involved in protecting convoys in the North Atlantic.
Alan Turing, Joan Clarke and Stewart Menzies discuss in a cafe the breaking of the Enigma code and how to keep information limited. However, there is a couple also in the cafe who could have overheard the Enigma conversation. Hardly the ideal situation to discuss top secret information.
After Alan's realisation of how to crack the code, they all rush back to the hut and begin rifling through the day's intercepted messages. Joan pulls one out and reads it in English. If they had yet to crack the code, she shouldn't have been able to read the message. The film didn't make it clear that not every transmission sent by the German military was necessarily encoded using Enigma, and so Bletchley Park had access to many uncoded messages in order to help them with deciphering the coded ones.
When leaving Turing's home after investigating the break-in, the sergeant, wearing helmet and cape, starts to get into the driving seat of a large black police car; he would have arrived either on foot or on a bicycle and would certainly not have driven a car, especially wearing a helmet and cape.
Benedict Cumberbatch's depiction of Turing suggests he was on the autism spectrum and struggled to socialise - claiming to not know what a joke is. (Alex Lawther's portrayal of young Alan is even more so.) But while Turing was undeniably eccentric and preferred to work alone, his friends described him as funny, warm, and charming. Children liked him too.
Turing lists the Ardennes as one of the battles influenced by Enigma. In actuality that battle opened as a surprise to the Allies because the Germans did not use Enigma to transmit orders for it, relying instead on telephone communications which could not be tracked at the time.
In interviews conducted with many of the people who worked at Bletchley Park, the interviewees have said that they had been instructed not to ask about activities in other huts. There are several scenes where other characters do just that, such as when the code-breakers celebrate in the pub and ask Helen about her work in her hut, and when Joan mentions about how the girls in one hut had been in a row over events in another hut.
When the police are visiting Turing's home after the break-in, Nock introduces himself as 'Detective Nock'. This is an Americanism, and he would say Detective Constable Nock (or Det. Sergeant etc.)