By the time he was at university, Alan Turing had sufficient facility at reading German (in gothic type no less) that he was reading advanced texts on Quantum Mechanics. In chapter 4 of the biography "Prof: Alan Turing Decoded" in the section The domination of the Germans, Sir Dermot Turing notes
One of the prize books awarded to Alan by Sherborne was Mathematische Grundlagen der Quantenmechanik (Mathematical Foundations of Quantum Mechanics) by John von Neumann, whom we will encounter again. [...] Alan reported that von Neumann was 'very interesting reading, although the applied mathematicians seem to find it rather strong meat'.
It was strong meat because it was in German, a language in which Alan had demonstrated his lack of aptitude for languages at Sherborne; but if you were serious about maths in the 1930s, you had to be able to get on with mathematical German.
The book can be seen at the Bletchley Park museum and is clearly well-used, including tea stains next to important results.
Moreover, the cryptanalytic community at Bletchley Park adopted various cod-Germanic words in their jargon. Among these are Turingismus and Banburismus which are attributed to Turing.
I'm also told by classified historians that the tri-partite liaison meetings between the British, French and Polish cryptanalysts on Enigma (at some of which, Turing was present) were conducted in German as it was the common language that all of the cryptanalysts knew. This is confirmed in part by chapter 6 of Herivelismus by John Herivel.
Post-war there is an elusive story that Turing was part of a group that visited Germany to understand Konrad Zuse's progress in developing a computer. Accounts say that the "interrogation" took the form of a colloquium.
It's known that Turing could occasionally stutter, which might have limited his ability to demonstrate fluency in a secondary school classroom, but it is clear that his command of written German was very strong indeed.
There are many things to commend about The Imitation Game, but it is not intended to be a documentary.