Keays-Byrne was born in Srinagar, in the state Jammu and Kashmir in India, to British parents. His family moved to Britain when he was a young child. He was schooled in England and began his career as a stage actor. Between 1968 and 1972, he had parts in Royal Shakespeare Company productions including As You Like It, The Balcony, Doctor Faustus, Hamlet, King Lear, The Man of Mode, Much Ado About Nothing, The Tempest or The Enchanted Island and Troilus and Cressida.