Davenport was born in Great Shelford, Cambridgeshire, to Katherine Lucy (née Meiklejohn) and Arthur Henry Davenport. His father was a bursar at Sidney Sussex College, Cambridge. He grew up in an academic family and was educated at St Peter’s School, Seaford, Cheltenham College and Trinity College, Oxford. Originally he chose to study Philosophy, Politics and Economics but switched to English on the advice of one of his tutors.
Davenport first appeared on stage at the Savoy Theatre and then with the Shakespeare Memorial Company, before joining the English Stage Company, one of its earliest members, at the Royal Court Theatre in 1956. He began appearing in British film and television productions in supporting roles, including a walk-on in Tony Richardson’s film, Look Back in Anger (1959). Subsequent roles included a theatre manager opposite Laurence Olivier in the film version of The Entertainer and a policeman in Michael Powell’s Peeping Tom (both 1960).
He made an impression as Thomas Howard, 3rd Duke of Norfolk in A Man for All Seasons (1966), costarred with Michael Caine in the war movie Play Dirty, and had a major role as Lord Bothwell in Mary, Queen of Scots. In 1972, he appeared as George Adamson, opposite Susan Hampshire in Living Free, the sequel to Born Free.
During the production of Stanley Kubrick’s 2001: A Space Odyssey, Davenport read the lines of HAL 9000 off-camera during the computer’s dialogues with actors Keir Dullea and Gary Lockwood. Canadian actor Douglas Rain was ultimately chosen for the role of HAL’s voice. Davenport took the leading role in the off-beat Phase IV (1974), which failed to find an audience. In 1979 he portrayed King George III in Prince Regent.
In February 1997, Davenport was the subject of This Is Your Life when he was surprised by Michael Aspel at David Nicholson’s stables near Cheltenham.
He was president of Equity from 1986 to 1992.