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.

He was responsible for producing and commissioning programmes produced in-house at the BBC, of which the greatest successes include The Office and French & Saunders. Plowman became Head of Comedy in October 2005, and oversaw the BBC's in-house comedy production, but no longer commissioned programmes. In June 2007, Plowman announced he was quitting his post at the BBC after 27 years. He decided to become a freelance producer for other shows and hoped to carry on his relationship with the BBC, continuing to create programmes "for them and elsewhere." In December 2003, The Observer named him in its list of the '50 Funniest or Most Influential People in British Comedy'. On 14 March 2006, he was honoured with the 'Judges' Award for Outstanding Contribution to British Television' at the Royal Television Society awards. Later he moved into the world of theatre. He co-produced Lucky You, the Carl Hiaasen best-seller that premiered as a theatre production at the Edinburgh Festival Fringe in 2008.