His acting career began in 1945. He co-founded the Gateway Theatre in Edinburgh in 1953 before joining the Royal Shakespeare Company in 1962. That year he played the title role in William Gaskill's production of Cymbeline. In 1965, he founded a company at the Royal Lyceum Theatre in Edinburgh. He also became the director of The Scottish Theatre Company for most of its years in the 1980s. On television, he played the title role in the 1956 BBC children's series Jesus of Nazareth. In 1983, he played the part of Lord Reith, the BBC's first Director General, in a two-part BBC production written by Roger Milner, entitled simply Reith.
He was a commentator for the BBC telecast of the Edinburgh Military Tattoo from 1966 until 2008. He was the BBC commentator for the Eurovision Song Contest 1972 in Edinburgh. He was a commentator on BBC television coverage of state events, and provided commentary outside Westminster Abbey for the Coronation of Elizabeth II in 1953. He commentated on the annual National Service of Remembrance at the Cenotaph between 1966 and 1988, as well as royal weddings and funerals, for example the state funerals of Princess Diana and the Queen Elizabeth The Queen Mother.
Fleming, who never married, was organist, lay preacher, secretary and reader at the Canonmills Baptist church in Edinburgh. After a long illness he died in St Columba's Hospice in Edinburgh on the night of Sunday 18 April 2010, aged 82.