Bowman began singing as an alto chorister in the choirs of Ely Cathedral and Oxford's New College, where he received degrees in education and history. His solo career began with a successful audition for Benjamin Britten's English Opera Group in 1967. The composer offered him the role of Oberon in his A Midsummer Night's Dream, as well as a concert at London's Queen Elizabeth Hall. Bowman continued to sing Oberon throughout his career, and made an excellent recording of the role for Hyperion in 1990. His association with Britten would continue, as well; the composer wrote the role of Apollo in Death in Venice for him, and also dedicated his Canticle No. 4 to Bowman. In 1970, Bowman became the first countertenor to sing at Glyndebourne, in a production of Cavalli's La calisto. By 1972, he had also debuted with the English National Opera and Covent Garden in operas of Handel (Semele) and Peter Maxwell Davies (Taverner). He would eventually sing most of the great Handelian operas, including Xerxes, Giulio Cesare, Orlando, and Rinaldo. Bowman has premiered a number of contemporary works, aside from those already mentioned. He was the first to sing the role of Astron in Michael Tippett's opera The Ice Break, in 1977; he sang Alan Ridout's Phaeton for BBC Radio; Michael Nyman wrote his Self-Laudatory Hymn of Inanna and Her Omnipotence for Bowman; and Bowman was the first to record John Tavener's Akathist of Thanksgiving and a number of works by Geoffrey Burgon. As an oratorio singer, Bowman has been successful in standard roles, like Handel's Messiah and Chandos Anthems, Vivaldi's Salve Regina, Orff's Carmina Burana, and Henry Purcell's Birthday Odes. Purcell's music has been a particular focus of his recordings on the Hyperion label, most of which have been in collaboration with Robert King and the King's Consort.
Paris has been almost a second home to Bowman during his career, and in recognition of his contribution to Parisian culture, the government made him Chevalier de l'Ordre des Arts et des Lettres in 1992. He also presented a sold-out 25th anniversary concert at the Palais Garnier that same year.