Osborne played violin in the school orchestra, but when he went to the Guildhall School of Music in London it was to study clarinet. He could play piano too, but after turning professional his main instrument was the alto saxophone. Osborne listed his favourite players as Phil Woods, Joe Henderson and Jackie McLean, with whom he shared a sharp-edged, slightly distressed-sounding tone, but he was often compared with Ornette Coleman for the urgency of his sound and his ability to create long, intense, graceful skeins of free melody.

Osborne first came to notice with the Mike Westbrook Concert Band when it re-formed after Westbrook’s move to London from Plymouth. ‘Ossie’ sat in for a couple of gigs and was asked to join permanently in early 1963, when he was one of only two professional musicians in the orchestra. Over the next few years Osborne was an important constituent of several fine bands, including the Michael Gibbs Band, Chris McGregor’s Brotherhood Of Breath, Harry Miller’s Isipingo and the group, ranging from a quartet to an octet, that he co-led with John Surman during 1968 and 1969. He also worked with John Warren, Alan Skidmore, Kenneth Terroade, Rik Colbeck, and Humphrey Lyttelton. In 1969, he established his own exceptionally exciting trio with Harry Miller and Louis Moholo, which, apart from many fine public gigs, recorded several superb sessions for BBC Radio’s Jazz Club. In the early 70s he began a fruitful association with Stan Tracey in wholly improvised duets, which brought Tracey back from the brink of retirement through disillusion with the music business. In 1973, Osborne co-founded S.O.S., probably the first regular all-saxophone band, with Skidmore and Surman. He was voted best alto saxophonist in the Melody Maker poll every year from 1969-73.

During the late 70s Osborne became increasingly ill and was unable to play in public from 1980 onwards. His relatively small recorded oeuvre does show the range of his playing, from deeply moving but unsentimental ballad interpretations with Westbrook to scorching, intense free explorations with Isipingo or the trio. He died from lung cancer in 2007.