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 #1.4 - S1-E4
Other mistake: A major plot point is the rivalry between the Wehrmacht and the SS, who refuse to work together. After the explosion at the cemetery, it is the Wehrmacht who are explicitly described as rounding up and interning suspected resistance fighters, with the SS not being party to it. However, when Archer visits the internment camp, most of the staff are wearing SS uniform.
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.
Continuity mistake: Archer is at various times described as chief superintendent (a rank which wasn't introduced until 1949), superintendent and inspector. Nobody, including him, seems to know what his actual rank is.
Episode #1.1 - S1-E1
Other mistake: The German officer in the pub is described in the credits as an unteroffizier (corporal or, generically, NCO). He actually wears the uniform of an oberleutnant (lieutenant), a commissioned officer.
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.