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.[8] 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).[8]
In the 1962 last episode of the first season of the TV series The Saint, titled "The Charitable Countess", with Roger Moore as Simon Templar and Patricia Donahue as Countess Rovagna, Davenport played a supporting role as the Countess's confidant, Aldo Petri. Davenport appeared again in The Saint in season 3, episode 16 (titled "The Rhine Maiden") as Charles Voyson.
During the production of Stanley Kubrick's 1968 film 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. However, Kubrick thought that Davenport's English accent was too distracting and dismissed him after a few weeks. Canadian actor Douglas Rain was ultimately chosen for the role.[10] 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.[1]
He appeared as Ebenezer Scrooge's grudging father Silas in the George C. Scott version of A Christmas Carol (1984), and played opposite Michael Caine again in the 1988 Sherlock Holmes spoof Without A Clue, which was Davenport's second-last feature film.