Branscombe's father and grandfather were also professional musicians. He played drums with Victor Feldman in a talent show as a child. He began on alto sax at age six, and played in the army with Jeff Clyne in 1954-56. He toured and recorded with Vic Ash in 1958, recorded with Tony Kinsey in 1959, and toured Japan with Stanley Black in 1960. He worked with John Dankworth as pianist and vibraphonist intermittently between 1960 and 1972, including in the 1963 film The Savant. He joined Harry South's band at Ronnie Scott's club in the mid-1960s, and played as a sideman with Tubby Hayes (1964), Stan Tracey (1966–68), Paul Gonsalves (1969), Ben Webster (1965, 1970), and Albert Nicholas (1973). He toured in Europe with Stan Getz in 1970, and played with the Lamb-Premru group around 1971. He also recorded as a leader with Kinsey and Tony Coe as sidemen on the album The Day I Met the Blues (EMI, 1977). As a session musician, he played tenor saxophone on The Beatles "Got to Get You into My Life"
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With Kenny Wheeler
- Song for Someone (Incus, 1973)
- Skrimshire/Kernfeld, "Alan Branscombe". Grove Jazz online.