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).