After moving to London in 1969, he joined the band Delivery,[2] one of the side roots of the Canterbury scene with Phil Miller, Pip Pyle and Lol Coxhill.

[2] He contributed to albums by Alexis Korner, Mike d'Abo, Chris Spedding, folk singers Harvey Andrews and Schunge, and was a part-time member of the bands Solid Gold Cadillac (jazz pianist's Mike Westbrook rock band) and Keith Tippett's Ovary Lodge.

Babbington's funk- and rock-oriented electric bass playing went along well with Karl Jenkins' and John Marshall's fusion concept of Soft Machine at the time.

After leaving Soft Machine, Babbington remained active on the UK jazz scene, playing with Barbara Thompson's Paraphernalia, Joe Gallivan's Intercontinental Express and various bands led by pianist Stan Tracey.

On 7 December 2021 Soft Machine issued a press release announcing that Babbington was retiring from the band, and was replaced by Fred Thelonious Baker on bass.