McBurney was born in Stockbridge, Massachusetts, the son of Dorothy Lillian (Rundall) and Henry McBurney, and the grandson of Charles McBurney, the American surgeon (known to subsequent generations of surgeons for defining McBurney's point). His mother was English, the daughter and granddaughter of British Army officers. He spent his childhood in the USA and then in Switzerland.
In 1933, he entered King's College, Cambridge, reading French and German, and then Archaeology and Anthropology. Graduate studies were interrupted by war service in the RAF. He completed his PhD (a study of European flint assemblages) in the years immediately following World War 2.
In 1952, he became a lecturer in archaeology at Cambridge, and later Reader and finally Professor of Quaternary Prehistory. His work included studies of the Upper Palaeolothic in Britain, important excavations in the Channel Islands, extensive excavations in Libya (the Haua Fteah cave) and, in later years, excavations in Iran and Afghanistan. He also published on French prehistory, archaeological work in the Soviet Union, and on cave art. His continuing influence is felt in the work of his many distinguished pupils.