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Elsdon studied trumpet under Tommy McQuater among others. His early professional work included time with Cy Laurie (1955–57), Graham Stewart (1957–59), a Royal Air Force band, and Terry Lightfoot (1959–61); with Lightfoot he played alongside Kid Ory and Red Allen. Elsdon led his own band from 1961 into the 1980s, and during the 1960s also played with Edmond Hall, Albert Nicholas, Wingy Manone, and Howlin' Wolf. He played in Keith Nichols's Midnight Follies from 1978 to 1985, as well as in small groups with Nichols around the same time. From the 1980s he was also active as a writer on music and educator.
- Jazz Journeymen (Black Lion Records, 1977)
- Dixieland Favourites (EMI & Hamlyn Records, 1970)
- Alan Elsdon Presents... (Columbia Records UK, 1964)
- Digby Fairweather, "Alan Elsdon". The New Grove Dictionary of Jazz.