Wright was educated at Chetham's School of Music, Manchester, and played the cello as a youth. He studied music at Royal Holloway College, University of London, and earned a B.Mus. in 1977. On graduation, he took a sabbatical year, 1977-78, as the elected President of the Student Union.
From 1978 to 1987, Wright worked at the British Music Information Centre (BMIC), as librarian and manager, then as director. He served as a senior producer for the BBC Symphony Orchestra from 1987 to 1989. He became the artistic administrator of The Cleveland Orchestra in 1989. He left his Cleveland post in 1992 for Deutsche Grammophon (DG), where he became an executive director and vice-president, and worked there until 1997.
In March 1997, Wright took up the newly created BBC post of Head of Classical Music, in charge of the BBC's orchestras, choirs, and bands. In 1998, he became Controller of Radio 3. During his Radio 3 tenure he raised the profile of jazz and world music, causing controversy among listeners. Other Radio 3 programming changes such as a perceived diminution of live music broadcasts have also attracted controversy., in addition to a perceived dilution of the level of programming. Wright was named Director of the BBC Proms in April 2007 and formally took up the post in October 2007, succeeding Nicholas Kenyon. In March 2014, he announced his resignation from the BBC to become chief executive at Aldeburgh Music, effective September 2014. He formally stood down as Controller of Radio 3 and of The Proms in July 2014.
In 2002, Wright was awarded an Honorary Fellowship of Royal Holloway College. He is also an Honorary Fellow of the Royal College of Music and a Fellow of the Radio Academy. His publications include the volume New Music 1989, in collaboration with Michael Finnissy.
Wright and his wife Rosie, a yoga teacher, have two children, Alice and William.