Bowen was born in Oxford, son of the chemist Edmund Bowen. 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.
His first post was with the Atomic Energy Research Establishment (AERE), working at the Wantage Research Laboratory, then in Berkshire. His early work started an interest in radioisotopes and trace elements that he maintained throughout his working life. While at AERE, he spent several months in 1956 attending the British nuclear tests at Maralinga in Australia to study the environmental effects of radiation.
Bowen realized that the calibration of different instruments intended to measure trace elements was an important issue that needed addressing. His solution was to produce a good supply of a material which later become known as Bowen's Kale. With Peter Cawse, he grew a large amount of the plant kale, then dried and crushed it into a homogeneous and stable substance that he then freely distributed to researchers around the world for years to come. This was probably the first successful example of such a standard.
In 1964, he was appointed as a lecturer in the chemistry department at the University of Reading. Later he was promoted to Reader in Analytical Chemistry in 1974. At Reading, Bowen undertook consultancy for Dunlop, investigating potential uses for their products. When the Torrey Canyon oil disaster occurred in 1967, he realized that it might be possible to use foam booms to block the oil from spreading in the English Channel. His original experiments were conducted in a small bucket in his laboratory. Although not entirely successful in reality at the time due to the rough seas, this lateral thinking combined his interest in chemistry with his love of nature and has since been effectively deployed to protect ports and harbours against encroaching oil slicks. Bowen wrote a number of professional books in the field of chemistry, including two editions of Trace Elements in Biochemistry (1966 and 1976).
From 1951 onwards, Bowen was a long-serving member of the Botanical Society of the British Isles (BSBI). He was meetings secretary for a period and the official recorder of plants for the counties of Berkshire and Dorset, producing Floras for both counties. He was also one of the leading contributors of botanical data for the Flora of Oxfordshire.