Denham was born in Beckenham, the son of Eleanor Winifred (née Lillico) and Norman Denham. He was educated at Tonbridge School and trained as a lift 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. Like fellow actor James Robertson Justice he played amateur rugby for Beckenham RFC.

In 1936 he married Elizabeth Dunn, with whom he had two sons and a daughter. She died in 1971. He was awarded the OBE in 1992.

Among his television appearances were as the father in Talking to a Stranger (1966), The Lotus Eaters (1972–73), as Archbishop Lang in Edward & Mrs Simpson (1978), Gerrit Dou in Schalcken the Painter (1979), All Passion Spent with Dame Wendy Hiller (1986), as Mr. Justice Gwent-Evans in an episode of Rumpole of the Bailey (1987), Behaving Badly (1989), Inspector Morse (1991) and the Sherlock Holmes story The Last Vampyre (1993).

In his book British Film Character Actors (1982), Terence Pettigrew noted that Denham 'had one of the best-known bald heads in British films. His face was a minor work of art, a bright-eyed pixie face hand-painted on an egg. It could be kindly, sympathetic, gnomish and infinitely expressive. He also had one of the most listenable and controlled of English-speaking voices, a legacy from his many years in radio. He died in 2002, aged 92 at Denville Hall in London.

He made a guest appearance in the BBC science fiction television series Doctor Who in the 1984 serial The Twin Dilemma, the first story to star Colin Baker in the title role as the Sixth Doctor. He later appeared in the Doctor Who radio serial The Paradise of Death in 1993 alongside Jon Pertwee. As The Honourable Mr Justice Stephen Rawley in two episodes of the BBC prison comedy Porridge, he ends up sharing a cell with Fletcher, whom he had sentenced.

Denham initially made his name in radio comedy series such as ITMA and Much Binding in the Marsh (providing over sixty different voices, female as well as male, according to a radio interview in November 1988), and later provided all the voices for the animated version of Animal Farm (1954). He was nominated for the BAFTA Award for Best Actor in a Leading Role for his performance as Blore in 1954's The Purple Plain. Other film credits include 23 Paces to Baker Street (1956), Night of the Demon (1957), Two-Way Stretch (1960), Sink the Bismarck! (1960), H.M.S. Defiant (1962), Those Magnificent Men in Their Flying Machines (1965), The Day of the Jackal (1973), Minder on the Orient Express (1985) and 84 Charing Cross Road (1987).

In further radio work, he starred in a BBC Radio 4 version of the Oldest Member, based on stories by P.G. Wodehouse, from 1994 to 1999, as Rumpole in Rumpole: The Splendours and Miseries of an Old Bailey Hack, as Dr. Alex Andre Manette in A Tale of Two Cities, as 'Father' in Peter Tinniswood's Winston series, and also as Chief Inspector Jules Maigret in several series beginning in 1976. He also portrayed Hercule Poirot in a BBC radio dramatisation of The Mystery of the Blue Train.