He was born in Lanark the eldest son of William Elliot, of the livestock auctioneers' firm Lawrie and Symington, and his wife, Ellen Elizabeth Shiels. His mother died during the birth of his youngest sibling. The children were thereafter raised by the mother's relatives in Glasgow. They appear to have had a company, Shiels, Elliot and Nelson, who made farming equipment including the Shiels patent milking machine. Elliot was raised in Glasgow and educated at the high school in Lanark and The Glasgow Academy. One of his friends from the Academy, through university and beyond, was the playwright Osborne Henry Mavor.