Bedford was born in London. He studied music at the Royal Academy of Music there under Lennox Berkeley, and later in Venice under Luigi Nono. In the late 1960s, he was engaged to orchestrate Kevin Ayers' album, Joy of a Toy, on which he also played keyboards. This led to him playing keyboards with Ayers' band, The Whole World.
Through Ayers, he met Mike Oldfield, then The Whole World's bass guitarist. In the 1970s he conducted and orchestrated Oldfield's Orchestral Tubular Bells album, an orchestral adaptation of the record that made the Virgin record label a serious player. Bedford subsequently made a number of records for Virgin, some using orchestral forces, others featuring Bedford's keyboard playing. He later went on to work with a wide variety of musicians, including Elvis Costello, Lol Coxhill, A-ha, Frankie Goes to Hollywood, Robert Wyatt, Madness and Billy Bragg.
All this time, Bedford was also writing avant garde classical works. One of his better known works in Star Clusters, Nebulae and Places in Devon (1971), for chorus and brass instruments. In With 100 Kazoos (1971), an instrumental ensemble is joined by the audience who are invited to play kazoos. He has combined skilled and non-skilled musicians in other works as well, with Seascapes (1986), for instance, combining a full symphony orchestra with school children, and Stories From The Dreamtime (1991) written for 40 deaf children and orchestra.
From 1968 to 1980, Bedford taught music in a number of London secondary schools, and he is noted for the large amount of educational music he has written for children. The musical notation he uses is often unconventional, frequently making use of graphic notation, thus opening his works up to be performed by children and others who cannot read conventional notation.
In general, Bedford's music has a tendency to harmonic stasis, the main interest instead being created by shifting timbres and textures. In his music for voice, he has set many texts by the poet Kenneth Patchen.
From 1969 to 1981, he was Composer in Residence at Queen's College, London, and in 1996 was appointed Composer in Association with the English Sinfonia. In 2001 he was appointed Chairman of the Performing Right Society, having previously been Deputy-Chairman.