Spence was born in Bombay, India but was sent back to Scotland to study. He attended George Watson's College in Edinburgh, then the architecture school of the city’s Heriot-Watt University, before completing his architectural studies at the Bartlett School of Architecture in London.
His first post was as an assistant in the London office of Sir Edwin Lutyens (whose work was to have a profound influence on Spence's style), where he worked on designs for the Viceroy's House in New Delhi, India. He subsequently joined the London office of Rowland Anderson & Paul, where he worked with Sir William Kininmonth, then returned to Edinburgh in 1930.
Spence served in the British Army from 1939-1945, reaching the rank of major.
During the war, Coventry’s Anglican Cathedral had been almost completely destroyed during enemy bombing. In 1944, Sir Giles Gilbert Scott submitted a design proposal to rebuild the cathedral but this was rejected by the Royal Fine Arts Commission. In 1950, a competition was launched to find the most suitable design from a British Commonwealth architect. Over 200 entries were received, but Spence's radical design was ultimately chosen. Work began in 1956 and the structure was completed in 1962. Spence was knighted in 1960 for his work at Coventry, and also served as President of the Royal Institute of British Architects (1958-1960).
From 1961 to 1968, Spence was Professor of Architecture at the Royal Academy. Sometimes compared with Robert Adam for his attention to detail, particularly in incorporating bespoke furniture and other elements into interior spaces, Spence died in 1976 in Eye, Suffolk and was buried at Thornham Parva, Suffolk.