 Postgate was born in Hendon, Middlesex, England, into the Postgate family, as the younger son of journalist and writer Raymond Postgate and his wife Daisy (née Lansbury), making him the cousin of actress Angela Lansbury and maternal grandson of Labour politician, and sometime leader, George Lansbury.His other grandfather was the Latin classicist John Percival Postgate.His brother was the microbiologist and writer John Postgate FRS.