Bedford was born in Morley, West Yorkshire and attended the Royal Academy of Dramatic Art in London from 1952–55. At RADA, he was in the same class as Albert Finney, Alan Bates and Peter O'Toole.
Primarily a stage actor, he was known for his English-speaking interpretations of the French playwright Molière, including Tony Award nominated performances in Tartuffe, The Molière Comedies, and The School for Wives. He was also well-known for his work with the Stratford Shakespeare Festival where he directed and act in plays like The Tempest, Measure for Measure, Twelfth Night, Richard III, Private Lives, Two Shakespearean Actors, London Assurance, Equus, Blithe Spirit, King Lear, Waiting for Gadot, The Lunatic, the Lover & the Poet, and The Importance of Being Earnest.
While appearing in films like Grand Prix, The Punch and Judy Man, The Holy Terror, Mr. St. Nick, and A Christmas Carol: The Musical; Bedford appeared on television shows like Edgar Wallace Mysteries, Coronet Blue, The Name of the Game, Nanny and the Professor, Alfred Hitchcock Presents, Murder, She Wrote, Cheers, The Equalizer, and Frasier.
Bedford died from cancer on January 13, 2016 in Santa Barbara, California. His remains were cremated.