Woolfenden was born in Ipswich and educated at Westminster Abbey Choir School, London, and Whitgift School, Croydon. He studied music at Christ's College in Cambridge and went on to study at the Guildhall School of Music and Drama. He joined the Royal Shakespeare Company in Stratford-upon-Avon in 1961 and was Head of Music from 1963 to 1998. He was Artistic Director of the Cambridge Festival from 1986 to 1991. In 1995 he was a founder director of the English Music Festival which became the Stratford on Avon Music Festival. He was the Chairman of the Denne Gilkes Memorial Fund, a charity which supports young musicians and actors. He was the founder of the publishing company, Ariel Music. Woolfenden married Jane Aldrick in 1962 and they had three sons.
Woolfenden composed many pieces for wind bands, chamber ensembles and orchestral works, many of which have been recorded.
He composed around 150 scores for the Royal Shakespeare Company. He worked with many major European theatre companies, including the Comédie-Française, Paris; the Burgtheater, Vienna; the Teatro di Stabile, Genoa; and the Norwegian National Theatre, Oslo.