Plowman was educated in Welwyn Garden City, Hertfordshire and at University College, Oxford, where was a member of the University College Players and made friends with others who went on to establish successful careers in comedy. One, Mel Smith, directed Plowman in a production of Rosencrantz and Guildenstern Are Dead. After Oxford, Plowman followed Smith to the Royal Court Theatre, where he met the director Lindsay Anderson. Plowman worked in theatre for a while, then joined Granada TV.