Robert Bostock was born in Lavington, Hampshire, England on 24 May 1902. He studied at Trinity College, Cambridge from where he graduated in 1923. He then decided to pursue a career in architecture. In the late 1920s or early 1930s he became a partner in the London-based architectural firm Unsworth & Goulder which was subsequently renamed Unsworth, Goulder & Bostock. In the 1940s he was a partner with Leonard T. Wilkins in the architectural firm Bostock & Wilkins. Hugh Workman joined them as a partner in 1957
Bostock was elected a Licentiate of the Royal Institute of British Architects (LRIBA) in 1933. His address was given as The Grey House, Hartley Wintney, Hampshire in 1939. He was in practice as an architect until at least the late 1960s. He died in Basingstoke, Hampshire on 2 March 1975.