Born Robert Lindsay Stevenson in Ilkeston, Derbyshire, the son of Norman and Joyce Stevenson, after leaving school, Lindsay enrolled in the drama department of a technical college in Nottingham, and intended to become a drama teacher. However, friends at Nottingham Playhouse encouraged him to apply to Royal Academy of Dramatic Art (RADA), and in 1968 he was accepted there with the aid of a government grant. After he graduated, he took a job as a dialect coach for a repertory company in Essex, and then joined a regional theatre group.

Lindsay first came to prominence as the cockney layabout Jakey Smith in ITV comedy series Get Some In!, and he appeared in the fourth series of the BBC sitcom The Good Life. He was then given the starring role as incompetent revolutionary Wolfie Smith in the BBC sitcom Citizen Smith.

He followed this with roles in a number of the BBC Television Shakespeare productions, including Benedick in Much Ado About Nothing, and as Edmund in King Lear opposite Lord Laurence Olivier in 1984. However it is notable he did take part in an obscure Radio 4 show, What Are You Talking About?, in the early 1980s between Television Shakespeare productions, in an attempt to establish a comedy career.

Lindsay became famous in the UK in his role as incompetent revolutionary Wolfie Smith in the BBC sitcom Citizen Smith. Earlier, a young Lindsay appeared on The Good Life in the 1977 series 4 episode entitled Our Speaker Today.

Lindsay's success on Broadway and in the West End led to the starring role in the film Bert Rigby, You're a Fool, although it was not a commercial success. However, he continued to enjoy success on television, and in 1991 played the leading role in Alan Bleasdale's dark comedy serial G.B.H., for which he won a BAFTA for his performance.

He also starred in the surreal Channel 4 sitcom Nightingales, and appeared in the films Fierce Creatures and Divorcing Jack. In 1998 he was cast in the recurring role of Captain Pellew in the ITV mini-series Hornblower, based on the novels of C.S. Forester.

He was also the original choice for the lead role in the drama Cracker, but turned the part down as he didn't want to become too associated with heavyweight, darker drama characters. He later appeared as Fagin in the 1999 ITV Oliver Twist miniseries.