Plater was born in Jarrow-on-Tyne, England, although his family moved to Hull when he was a young child. Jarrow was much publicised as a severely economically depressed area before the Second World War, and Plater has commented that returning to the area for holidays was a source of much bemusement for his family and friends. Plater trained as an architect at King's College, Newcastle (later the University of Newcastle), but only practised in the profession briefly, at a junior level. He commented later that it was soon after he had to fend off a herd of pigs from eating his tape measure whilst surveying a field that he left to pursue full-time writing. Plater stayed in the north of England for many years after he became prominent as a writer and lived in Hull.
He first made his mark as a scriptwriter for Z Cars. His subsequent credits include The Stars Look Down (1974), Oh No, It's Selwyn Froggitt!, The Journal of Vasilije Bogdanovic, Close the Coalhouse Door, Get Lost! (1981), The Beiderbecke Affair (1985), and its two sequels, Misterioso, Oliver's Travels, a 1980 adaptation of J.B. Priestley's The Good Companions for Yorkshire Television, an adaptation of George Orwell's Keep the Aspidistra Flying, Belonging and Peggy for You, which was nominated in 2001 for a Laurence Olivier Theatre Award. He has also contributed to the BBC series Dalziel and Pascoe, and he adapted Chris Mullin's novel A Very British Coup (1989) for television. He was the driving force behind the TV version of Flambards, which under his influence was slanted well to the political left of K. M. Peyton's original books. Jazz is a recurring motif through much of his work, often referenced explicitly as well as underpinning his story structures.
Plater served as president of the Writers' Guild of Great Britain from September 1991 until April 1995. He has received honorary degrees from the University of Hull and the Northumbria University in Newcastle. In the New Year's Honours List published 31 December 2004 he was created a Commander of the Order of the British Empire for services to Drama.
Plater currently lives in London with his wife Shirley Rubinstein. He has three grandchildren living in Yorkshire, and six in Newcastle as well as various others scattered around the UK, 16 in all.
He is a supporter of Hull City AFC.