Beresford was born in Wellington, Shropshire in England, studied at the University of York, and stayed in York after graduating, becoming involved in theatre as well as arranging various free improvisation concerts in the city. At this early stage, he was playing a wide variety of music, playing the Hammond Organ in a soul music covers band, featuring on Trevor Wishart's early work Journey Into Space (1973) and free improvising.
In 1974, Beresford moved to London, where he played in Derek Bailey's Company events and in the groups Alterations with David Toop, Terry Day and Peter Cusack, and the Three Pullovers with Nigel Coombes and Roger Smith. He was also a member with Gavin Bryars and Brian Eno of the Portsmouth Sinfonia.
Beresford has continued to play free improvisation with a number of prominent musicians, including Evan Parker, Lol Coxhill, John Zorn and Han Bennink. He has also worked with a number of popular musicians, including The Slits, Frank Chickens and The Flying Lizards.
During the 1970s Beresford was a co-founder and co-editor of the magazine Musics, which dealt mainly with free improvisation, whilst during the early 1980s he helped to set up the somewhat glossier publication Collusion, which had a wider musical remit, covering fields such as rap, heavy metal, classical music, film music, pop music as well as the avant garde and free improvisation. Along with David Toop, Beresford was also a prime mover of the London Musicians Collective.