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Cook was born at his parents' house, "Shearbridge", in Middle Warberry Road, Torquay, Devon. He was the only son and eldest of the three children of Alexander Edward "Alec" Cook (1906-1984), a colonial civil servant, and his wife Ethel Catherine Margaret, nee Mayo (1908-1994). He was educated at Radley College and Pembroke College, Cambridge, where he studied French and German. As a student, Cook initially intended to become a career diplomat like his father, but Britain "had run out of colonies", as he put it.

Cook played multiple roles on the 1977 concept album Consequences, written and produced by former 10cc members Kevin Godley and Lol Creme. A mixture of spoken comedy and progressive rock with an environmental subtext, Consequences started as a single that Godley and Creme planned to make to demonstrate their invention, an electric guitar effect called the Gizmo, which they developed in 10cc. The project grew into a triple LP boxed set. The comedy sections were originally intended to be performed by a cast including Spike Milligan and Peter Ustinov, but Godley and Creme eventually settled on Cook once they realised he could perform most parts himself.  The storyline centres on the impending divorce of ineffectual Englishman Walter Stapleton (Cook) and his French wife Lulu (Judy Huxtable). While meeting their lawyers - the bibulous Mr. Haig and overbearing Mr. Pepperman (both played by Cook) - the encroaching global catastrophe interrupts proceedings with bizarre and mysterious happenings, which seem to centre on Mr. Blint (Cook), a musician and composer living in the flat below Haig's office, to which it is connected by a large hole in the floor.  Although it has since developed a cult following due to Cook's presence, Consequences was released as punk was sweeping the UK and proved a resounding commercial failure, savaged by critics who found the music self-indulgent. The script and story have evident connections to Cook's own life - his then wife Judy Huxtable plays Walter's wife. Cook's struggles with alcohol are mirrored in Haig's drinking, and there is a parallel between the fictional divorce of Walter and Lulu and Cook's own divorce from his first wife. The voice and accent Cook used for the character of Stapleton are similar to those of Cook's Beyond the Fringe colleague, Alan Bennett, and a book on Cook's comedy, How Very Interesting, speculates that the characters Cook plays in Consequences are caricatures of the four Beyond The Fringe cast members - the alcoholic Haig represents Cook, the tremulous Stapleton is Bennett, the parodically Jewish Pepperman is Miller, and the pianist Blint represents Moore.
Peter Cook