Episode #1.5 - S1-E5
Factual error: The facility raided at the end is referred to as belonging to the Ministry of Defence. Although Churchill held the post of Minister of Defence, the ministry itself was not created until 1946.
Factual error: Although he is openly an SD officer, Huth does not wear the SD insignia on his cuff.
Episode #1.3 - S1-E3
Factual error: Dunn says he's researched at the British Library. It wasn't called that until 1973. Before then it was the British Museum Library and always referred to either as that or simply as the British Museum.
Factual error: Huth does not wear his Knight's Cross correctly. It should be suspended below the collar, not attached high on the collar as his is.
Factual error: Gruppenführer Springer wears the rank insignia of a Brigadeführer, the rank below.
Factual error: The constables working at Scotland Yard are wearing the divisional letter 'S'. 'S' Division covered North West London, including Hampstead, Hendon and Golders Green. Its men wouldn't have been anywhere near Scotland Yard.