Innes was born in Essex, and studied at the Norwich School of Fine Art, from which he was thrown out around 1963, allegedly for "spending all day playing music, instead of making things".
In the period 1962 to 1965, Innes and several other art school students started a band which was originally named The Bonzo Dog Dada Band after their interest in the art movement Dadaism, but which was soon renamed the Bonzo Dog Doo-Dah Band (Often shortened to The Bonzo Dog band). Innes, with Vivian Stanshall, wrote most of the band's songs, including "I'm the Urban Spaceman", their sole hit, and "Death Cab for Cutie" (which inspired a Group of the same name), which was featured in the Beatles' film Magical Mystery Tour.
In the 1970s, Innes joined with Eric Idle, of the Monty Python team, to create the television comedy series Rutland Weekend Television. This show spawned The Rutles (the "prefab four"), a Beatles parody band, in which Innes played the character of Ron Nasty, who was loosely based on John Lennon.
He appeared in Monty Python and the Holy Grail, playing a head-bashing monk and the leader of Sir Robin's minstrels, and in Terry Gilliam's Jabberwocky.
On BBC television, he performed songs and sketches in The Innes Book of Records, punning on the Guinness Book of Records. The series has not been repeated.
During the 1980s, Innes found a new, younger audience, when he played the role of the Wizard in the children's television series Puddle Lane.
He also voiced the 1980s Children's cartoon adventures of The Raggy Dolls, a motley collection of 'rejects' from a toy factory. The 65 episodes, for Yorkshire television included the characters: Sad Sack, Hi-Fi, Lucy, Dotty, Back-to-Front and Princess.