Chelsom was born in Blackpool, Lancashire, the son of antiques shop owners Kay and Reginald Chelsom. He studied at the Central School of Drama in London. He worked as an actor for the Royal Shakespeare Company as well as taking part in numerous film and television productions including A Woman of Substance which also included Jenny Seagrove and Deborah Kerr. While acting he developed a growing interest in writing and directing. From 1985-98 he ran the television film course at Central School of Drama and later taught at both the Actors' Institute and Cornell University.