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, née Mayo (1908–1994). He was educated at Radley College and Pembroke College, Cambridge, where he studied French and German.