He studied composition with Edwin Roxburgh and Simon Bainbridge at the Royal College of Music, and won the Mendelssohn Scholarship in 2000.
In 2007, Bedford became the first compositional recipient of a Paul Hamlyn Artists' Award since David Sawer in 1993.
2007 also saw him receive a nomination for the Royal Philharmonic Society large-scale composition prize, for his song cycle On Voit Tout En Aventure.
Bedford has written two operas, Seven Angels with librettist Glyn Maxwell, and Through His Teeth with David Harrower.
In 2018 Bastille Musique released a studio recording of Through His Teeth, performed by Opera Factory Freiburg.