Roxburgh was born in Liverpool. After playing oboe in the National Youth Orchestra, he won a double scholarship to study composition with Herbert Howells and oboe with Terence MacDonagh at the Royal College of Music. He also studied composition with Nadia Boulanger in Paris and Luigi Dallapicolla in Florence. His students have included Jonathan Lloyd, Daniel Giorgetti, Kenneth Hesketh, Dai Fujikura, Luke Bedford, Stephen Mark Barchan, David Warburton and Helen Grime.
After his studies he became principal oboist of the Sadler's Wells Opera and taught composition and conducting at the Royal College, where he founded the RCM's Twentieth Century Ensemble. Together with Leon Goossens he wrote the Menuhin Music Guide for the oboe in 1977.
In 2004, Roxburgh became the acting Head of Composition at the Birmingham Conservatoire and from 2005 has acted as visiting tutor in composition and conducting, as well as workshop leader. In 2007 his 70th birthday was celebrated in a series of concert performances showcasing a selection of his works. In 2008 he received the Royal Philharmonic Society Elgar Bursary.
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