Episode #1.1 - S1-E1
Factual error: When reviewing the phone numbers in the 1970s diary, they start '01' which was a change to UK numbers only brought in 1995.
Factual error: Both detective superintendents always wear uniform, which is only worn by detectives on very formal occasions if at all. Both also wear collar badges, which are not worn by either the Metropolitan Police or Thames Valley Police, the forces to which they belong.
Episode #4.2 - S4-E2
Factual error: Ram says he has two "Chief Constable's Commendations" (which are also shown on his office wall). He has spent his entire career in the Metropolitan Police, which has a Commissioner instead of a Chief Constable.