I would recommend 'Case Histories' by Kate Atkinson. It follows the investigations of a private detective named Jackson Brodie. The stories are interwoven in an interesting way. Then there's 'The Dry' by Jane Harper. It's set in Australia but written by a British - born author. It has a great mystery and a vivid sense of place. Also, 'The Seven Deaths of Evelyn Hardcastle' by Stuart Turton is quite unique. It has a time - loop concept that makes it very engaging.
Some popular ones are 'Tinker, Tailor, Soldier, Spy' by John le Carré. It's a classic of the genre with a complex spy - like detective story. 'Silent Witness' by Nigel McCrery is also good. It delves deep into forensic investigations. And 'The No. 1 Ladies' Detective Agency' by Alexander McCall Smith. It has a different take as it's set in Botswana but has a very British - influenced style of detective work and storytelling.