Lindup was credited as the composer of TV theme tunes such as The Informer, and occasional film and TV scores such as Games That Lovers Play (1971), The Persuaders! (1971), White Cargo (1973), Shatter (1974), The Box (1974), The Spiral Staircase (1975) and the film version of Rising Damp (1980). The library music he composed is usually uncredited in films and TV programmes, although a younger audience is likely to have heard it in recent years; Midnight Serenade is featured in two Electronic Arts PlayStation 2 video games (The Godfather and The Sims 2: Castaway). That song was also included in the 2018 film The Catcher Was a Spy. The animated series SpongeBob SquarePants is another beneficiary of Lindup's work. His most prominent pieces are orchestrations for musical films such as Scrooge (1970) and Goodbye, Mr. Chips (1969). The trailer of the film The Full Monty (1997) credits use of Lindup's composition The Zodiac.
His recording of the Dankworth-David Dearlove song "Let's Slip Away", accompanying Susan Grey, features as source music in the films Saturday Night and Sunday Morning (1960) and A Taste of Honey (1961).
Lindup was born in East Preston, West Sussex. He was married to Belizean-born British actress and singer Nadia Cattouse; their son Mike is the keyboardist for Jazz-pop band Level 42. Lindup died in Southampton on 7 January 1992.