From 1905 he studied science and medicine at the University of Glasgow. In the election for Rector of the University of Glasgow in 1908, Elliot assisted the Liberal Club campaign to elect the Liberal David Lloyd George, supported the candidacy of George Curzon, 1st Marquess Curzon of Kedleston, and according to Henry Mavor, ultimately voted for Keir Hardie, founder of the Labour Party. In 1909–10 he was editor of the "Glasgow University Magazine". He co-wrote the university song "Ygorra". Elliot graduated BSc in 1910. As a postgraduate he was President of the Glasgow University Union, 1911–12. While in the British Army's Glasgow University Officers' Training Corps, he befriended John Boyd Orr. He graduated MB ChB in 1913.