Tilbury studied piano at the Royal College of Music with Arthur Alexander and James Gibb and also with Zbigniew Drzewiecki in Warsaw.
During the 1960s, Tilbury was closely associated with the composer Cornelius Cardew, whose music he has interpreted and recorded and a member of the Scratch Orchestra.
Tilbury has also recorded the works of Howard Skempton and John White, among many others, and has also performed adaptations of the radio plays of Samuel Beckett.
With guitarist AMM bandmate Keith Rowe's electroacoustic ensemble M.I.M.E.O., Tilbury recorded The Hands of Caravaggio, inspired by the painter's The Taking of Christ (1602).
Critic Brian Olewnick describes the album as "A staggering achievement, one is tempted to call The Hands of Caravaggio the first great piano concerto of the 21st century".