His childhood was spent Swayne first in Singapore and Australia, and later in Liverpool and Yorkshire. He began assisted by his cousin Elizabeth Maconchy, at the age of twelve years with composing. After his schooling at Ampleforth College, he studied at the University of Cambridge, where he was instructed by Raymond Leppard and Nicholas Maw.
In 1968 he received a scholarship to the Royal Academy of Music. There he had Nicholas Maw, Harrison Birtwistle and Alan Bush as a teacher. In the years 1976 and 1977 he attended classes with Olivier Messiaen at the Paris Conservatoire. A study trip led him 1981/1982 to Gambia and Senegal. With his second wife, a native Ghanaian, he spent the years 1990 to 1996 in eastern Ghana, before he went back to London, where he still lives and works today.
Swayne teaches composition at the University of Cambridge and is a research associate and Composer in Residence at the local Clare College.
- CRY, op dedicated 27 for 28 singers and electronic instruments, commissioned by the BBC, Olivier Messiaen, written in 1980
- Magnificat, Op 33, for eight -part choir a cappella, commissioned by the Christchurch, composed in 1982
- String quartets 1-3, Op 8, 24, 61, composed in 1971, 1977 and 1993
- HAVOC, op 83, continuation of CRY, composed in 1999