Other mistake: During the opening sequence. Sadie Buckingham walks out of Smithfield market, walks down St John's Street, Cawcross Street, then up Greenhouse Rents. This is a dead end. She next appears walking along Jerome street where she is murdered. After the titles DCI Luther walks down through the police cordon and meets Martin. In the background is Godfrey Phillips Ltd, which is a company at the junction of Commercial Street and Jerome Street. This is a 30 minute walk from Smithfield. Luther then says "is he having a joke... slaughtered like cattle next to a meat market".
Other mistake: A detective is watching footage on a computer screen of the standoff with Luther and Reed where police snipers are on rooftops. Someone walks across the screen right behind Luther and Reed. Surely the police would have cordoned off the area with the snipers on the rooftops.