Kelly was educated at Downside School, a Catholic independent school in Stratton-on-the-Fosse in Somerset, followed by the University of Cambridge.

Kelly is probably best known as the presenter of Wish You Were Here...?, Food and Drink and Clapperboard. He is also the former owner of the Midsummer House restaurant in Cambridge.

Kelly worked as a continuity announcer for Anglia Television in 1963 before commencing a long career as a presenter.

He has presented episodes of: Sixth Form Challenge, Kellyvision, The John Thaw Story, Vintage Quiz, Food and Drink, Clapperboard, Wish You Were Here...?, Anything You Can Do , Eat Drink and be Silly, Junior Criss Cross Quiz, Quisine and was one of the off-screen narrators of World in Action.

His career as a TV producer has included Life After Tomorrow, Days That Shook the World, The Great Train Robbery, A Line in the Sand, Holiday Showdown, Pornography: The Musical, The Heat Is On, Monsignor Renard, Without Motive, Castaway 2000, Kavanagh QC, Memento Mori, Diplomatic Baggage, Ancient History, Job Satisfaction, A Sense of Loss, Soldier Soldier, The Last Post, Lost and Found, Saving Face, Lifelines and A Touch of the Sun.

He has written episodes of Soldier Soldier, Minder, Lovejoy, El C.I.D. and Saracen.