Kate Atkinson's 'Case Histories' is a great modern British mystery. It weaves together three different cases that the detective, Jackson Brodie, has to solve. Atkinson's writing is sharp and her characters are vivid. The novel gives a great sense of the modern British landscape while also providing a complex and engaging mystery.
Sure. 'The Cuckoo's Calling' by Robert Galbraith (J.K. Rowling). It has a great detective, Cormoran Strike, and the mystery is well - crafted. The story is set in London and deals with the seedy underbelly of the city and the world of fashion.
Another good one is 'Gone Girl' by Gillian Flynn. Although Flynn is American, the novel has a very British - style mystery feel to it. The story is full of mind - games between the two main characters, and the plot twists are shocking. It's a real page - turner that keeps you on the edge of your seat.