Bowen was born in Oxford, son of the chemist Edmund Bowen and Edith Bowen (nee Moule). He attended the Dragon School, gaining a scholarship to Rugby School and then a demyship to Magdalen College, Oxford. He won the Gibbs Prize in 1949 and completed a DPhil in chemistry at Oxford University in 1953 before starting his professional career as a chemist. Bowen was also a proficient amateur actor in his early years, appearing with a young Ronnie Barker at Oxford.