Drake was born in London on April 5, 1959. He began playing piano at seven and took lessons at the Purcell School. Moving on to the Royal Academy of Music, he made his debut at the Purcell Room in London in 1981. Early in his career, he specialized in the music of Schumann. Drake has recorded extensively since the late 1990s. A noted early release, with soprano Katarina Karnéus, featured songs by Jean Sibelius. A versatile vocal accompanist, Drake has often mastered unfamiliar repertory such as the Haydn Canzonettas (with Christoph Genz), African American spirituals (with Ruby Philogene), and songs by Ivor Gurney (with Paul Agnew). His most frequent vocal collaborator has been tenor Ian Bostridge, but he has also worked with Matthias Goerne, Lorraine Hunt Lieberson, Victoria de los Angeles, and many other distinctive vocal stylists. His instrumental collaborators include clarinetist Emma Johnson and hornist Richard Watkins, whom he accompanied on the 2019 release The Romantic Horn (Signum Classics). Also in 2019, Drake backed tenor Nicky Spence on a recording of Janacek's rarely heard The Diary of One Who Disappeared and Nursery Rhymes. During this time, Drake was in the process of recording the complete songs of Liszt with soprano Anna Prohaska.
Drake has served as the director of the Perth International Chamber Music Festival in Australia and the Machynlleth Festival in Wales. He has devised song series for Wigmore Hall in London, the BBC, and the Royal Concertgebouw in Amsterdam. Drake teaches song accompaniment at Graz University for Music and Performing Arts in Austria and has given numerous masterclasses.