Burnap played washboard in a skiffle band with classmates as a teenager in England. He moved to New Zealand at age 19, where he began playing trombone and played in the Omega Jazz Band in 1960-62. From 1962 to 1965 he played in Australia with the Hot Sands Jazz Band (1962-64) and Geoff Bull's Olympia Jazz Band (1964-65). In 1965 he played for a time at Preservation Hall in New Orleans, then returned briefly to the UK, playing with Terry Lightfoot and Monty Sunshine. He spent three further years (1966-69) in Australia before moving permanently back to the UK, settling in London. He played with Ian Armit (1969-70), Alan Elsdon (1970-75), Alex Welsh (1978-79), and Acker Bilk (1980-87). After 1988 Burnap hosted jazz radio programs for the BBC and Jazz FM.