The punk band that recorded “Neat Neat Neat” on the 1977 album Damned Damned Damned was The Damned.
“Neat Neat Neat” is one of the best-known tracks on the band's debut album, which was produced by Nick Lowe. The record also includes “New Rose,” a single widely recognized as the first UK punk single, but this question concerns a different song from the same album.
The Damned formed in London in 1976 with Dave Vanian, Captain Sensible, Rat Scabies, and Brian James. Their debut was released in February 1977 and helped establish the speed and attitude of British punk. The band later moved into gothic rock and other styles, making its broader career more varied than its first album suggests.