Born in Bath, England, Lynn studied Law at the University of Cambridge (where he participated in the Footlights revue, but by the 1960s was appearing in and writing television sitcoms.
As an actor, his most memorable role was as Beryl's boyfriend in early series of The Liver Birds. As a writer, he worked on On the Buses and wrote for Harry Worth before hitting the big time, in partnership with Antony Jay, as writer of the Yes, Minister and Yes, Prime Minister series. He has also written films such as Nuns on the Run.