Assembling a quartet from Kent, England, comprising fellow Prisoner bass player Alan Crockford and ex-Daggermen personnel Simon Howard (drums) and Taylor's brother David (guitar), the band recorded a BBC session for disc jockey John Peel before Taylor retired to Sweden for a break. However, the broadcast made such an impression that the band was signed to new "mod" label Re-Elect the President. A mini-album of cover versions, Mission Impossible, featured "organ groovy" '60s soundtrack instrumentals like the single "Blow Up," with Jimmy Smith and Booker T. & the MG's providing the strongest influences. The Money Spyder took the theme a stage further; while the Damned had mocked the psychedelic soundtrack as Naz Nomad & the Nightmares, the JTQ reminisced on the beat and jazz age. Taylor become frustrated with the band's limitations, and by the time Wait a Minute appeared on Polydor Records dance offshoot, Urban, in September 1988, only his brother remained with him in the group.

For a powerful remake of "The Theme from Starsky and Hutch," new jazz musicians and ex-James Brown horn players were recruited, as the JTQ found themselves central to a new, London-based acid jazz movement. Howard and Crockford, meanwhile, provided the rhythm section for ex-Prisoners guitarist Graham Day's new project, the Prime Movers. A further development for the JTQ came with the recruitment of two rappers for May 1989's "Breakout." This single hinted at a move away from jazz toward the dance charts, but Do Your Own Thing combined both elements, alongside a continuing debt to the original fusion of jazz/dance and rare groove, not least on their rousing rendition of the '70s club favorite, "Got to Get Your Own." A long overdue live album, Absolute, was released in 1991 on the Polydor subsidiary, Big Life. While ex-Style Council and Jazz Renegades drummer Steve White served in the JTQ for a time, Taylor himself has also made several guest performances, including appearances for the Wonder Stuff, the Pogues, and U2. More permanent members of the JTQ include Gary Crockett (bass), Neil Robinson (drums), Dominic Glover (trumpet), and John Wilmott (saxophone, flute). "Love the Life" and "See a Brighter Day," featuring new lead singer Noel McKoy, saw them bid for chart success in 1993, but once the spurious acid house bubble had burst, Taylor was able to concentrate on making music without the pressure of following a trend. The band's subsequent work, which has been released on a variety of labels including Acid Jazz Records, JSP, and Gut, has seen a return to the Hammond groove-jazz style of Jimmy McGriff and Jimmy Smith. In 1997, Taylor's composition "Austin's Theme" was featured in the hit movie Austin Powers: International Man of Mystery.