Together they devised and wrote the BBC sitcom Hancock's Half Hour (1954–1961), the first two series of Comedy Playhouse (1961–1963), and Steptoe and Son (1962–1974).

Simpson was born in Brixton, South London, and was educated at Mitcham County Grammar School for Boys.

He was a football fan and supported Brentford and Hampton & Richmond Borough where Simpson was honorary club president.

He contracted tuberculosis aged 17 in 1947 and was admitted to Milford Sanatorium near Godalming in Surrey, where he spent 13 months.

[3] One of the plotlines in Linda Grant's 2016 novel The Dark Circle was based on Simpson's experience of broadcasting on hospital radio with Galton during their time at the sanatorium and its role in their subsequent careers.