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.
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.