Winsor was born on 7 December 1957 in Broughty Ferry, Dundee to Thomas V M Winsor and Phyllis Bonsor. He was educated at Grove Academy state comprehensive school in Broughty Ferry. In 1976, he went to the University of Edinburgh to study law. After graduation in 1979, he served his two-year Scots legal apprenticeship with Dundee law firms Thorntons & Dickies. He practised for a year after that doing mainly litigation in Dundee Sheriff Court. In 1982 he enrolled as a postgraduate student at the Centre for Petroleum and Mineral Law Studies of the University of Dundee under Professor T C Daintith. He received a Diploma in Petroleum Law in 1983. He became a Writer to the Signet (WS) in 1984.