After Steve Harley disbanded Cockney Rebel in 1977, Elliott was called by Andrew Powell to play drums on Kate Bush's first album, The Kick Inside.
As well as continuing session drumming for Kate Bush and The Alan Parsons Project he went to on to play for Justin Hayward's song "Suitcase", from the album Night Flight in 1980.
That same year he and the rest of The Alan Parsons Project core musicians team, Ian Bairnson and David Paton, were joined by the singer Colin Blunstone and Camel's keyboardist Peter Bardens, and formed Keats.
Elliott not only played drums and percussion, he also took part in the songwriting and, with Colin Blunstone, he wrote "Tragedy" and "Night Full of Voices".
Elliott appeared on various TV shows including Top of the Pops playing with Tina Turner, Eric Clapton, Roxy Music and Kate Bush.
[15] With Alan Parsons, Ian Bairnson and other different musicians and singers, he toured from 1994 to 2002 covering the US, South America, Japan and big part of Europe.
During this period he played at the Night of the Proms in 1997 as part of the Alan Parsons performance and with Simple Minds, Deborah Harry and Paul Young.