Born in Leith, Heatly was educated at Leith Academy and studied engineering at the University of Edinburgh. He began his diving career with Portobello Amateur Swimming Club and was self-taught. Heatly dominated diving in Scotland for 21 years, becoming the East of Scotland Champion (1937–39) and Scottish Champion (from 1946 - 1958). Having represented his country around the world, the highlight of Heatly's diving career was winning British Empire and Commonwealth Games gold medals for Scotland for 3 metre springboard diving (1954) and 10 metre highboard diving (1950 and 1958). He also won a silver medal in 1950 for 3 metre springboard diving, and a bronze medal in 1954 for 10 metre highboard diving.
Turning to a career in sports management, Heatly acted as Chairman of the Scottish Sports Council, (1975-1987) and Chairman of the Commonwealth Games Federation from 1982 to 1990, including the return of the Games to Edinburgh in 1986. Heatly was awarded a CBE in 1971 and knighted on 1990. He was also created a Deputy Lieutenant of the City of Edinburgh in 1984. He holds honorary degrees from the University of Edinburgh (1992) and Queen Margaret's College (1994).
Heatly had four children Anne, Jane, Peter and Robert, by his first wife who died in 1979. In 1984 he married Mae Cochrane who died in 2003. As of 2014 he lives in Balerno. Heatly was inducted as one of the first members of the Scottish Sports Hall of Fame in 2002.