Denham was born in Beckenham, Kent, the son of Eleanor Winifred (née Lillico) and Norman Denham. He was educated at Tonbridge School and trained as an elevator engineer. Denham eventually became an actor in 1934 and appeared in live television broadcasts as early as 1938, continuing to perform in that medium until 1997.
In addition to his early comedy roles, Denham is notable for his long association with Peter Tinniswood, playing "Father" in all the various "Winston" series. He starred in the acclaimed productions of Beyond the Pale by William Trevor, and A Slight Ache by Harold Pinter. He also played Inspector Maigret in several series beginning in 1976.