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.
Plot hole: The car was tracked down by finding the original car key on Jimmy's body. If he had stolen the car from the original owner, how did he come to have the original car key?
Continuity mistake: The series doesn't seem sure what Sunny's rank is. He is described at various times both on screen and in the credits as both Detective Inspector and Detective Sergeant.
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.