Lindsay was born in Edinburgh in 1906, son of George Herbert Lindsay, distiller and baillie (town councillor), and Helen Eliza Turnbull. He was educated at Marlborough College and Trinity College, Cambridge. Here he began a lifetime friendship with John Betjeman.
Like many artistically-minded Cambridge undergraduates of his generation, Lindsay came under the spell of "Manny": Mansfield Duval Forbes. In his circle, Lindsay made a number of friends who were to have considerable influence on his later work; amongst these were Raymond McGrath, Oliver Hill, Robert Hurd, Thomas Steuart Fothringham and Robert Simpson.
After leaving Cambridge he was apprenticed to Reginald Fairlie in 1927. In 1931 he commenced practice on his own account before joining the firm of Orphoot and Whiting in 1933. In 1932 Lindsay married the Hon Maysie Elizabeth Loch, daughter of Major General the 2nd Baron Loch of Drylaw and Stoke College.
During the 1930s Lindsay quickly developed a wide circle of personal and professional friends, many of whom were later to provide work for his architectural practice. Amongst these were the 4th Marquess of Bute, and his nephew Major Michael Crichton Stuart, Dr Francis Carolus Eeles (secretary of the Council for the Care of Churches), J S Richardson (principal inspector of Ancient Monuments) and Peter F. Anson the writer and historian.
The Dictionary of Scottish Architects states that Lindsay's circle of influential contacts was further widened when his sister, Ailsa Margaret Lindsay, married Lt Col Charles Findlay DSO, younger son of architect Lt Col James Leslie Findlay and grandson of John Ritchie Findlay of The Scotsman. The connection to the Marquess of Bute (and his links to the National Trust for Scotland) led Lindsay to be commissioned in 1936 to draw up lists of important buildings in 103 Scottish towns and villages, based on an Amsterdam model of three categories (A,B and C). Lindsay continued this work for many years, although interrupted by the outbreak of war, and this produced the basic list upon which statutory protection for listed buildings was later introduced in 1947, following expansion to cover all areas.