Cooper was born at Westbury-on-Trym, near Bristol, England. He was educated at Clifton College, and then at Keble College, Oxford, where he was an organ scholar. During the 1930s he worked initially as a church organist and piano teacher before joining the GPO Film Unit, where he wrote incidental music for documentaries, including Mony a Pickle (1938) and A Midsummer Day's Work (1939). Here his colleagues included the poet W. H. Auden and the composer Benjamin Britten. He had already embarked on a promising career as a concert pianist when the outbreak of World War II forced him to give up the concert platform for the duration of hostilities. He resumed his career in 1946, studying briefly with Egon Petri and making his London debut in 1947. As a concert pianist, Cooper made a number of successful recordings (including some for the World Record Club), and also began broadcasting on radio. In 1946 he assisted Ralph Vaughan Williams in arranging the latter's Piano Concerto for two pianos.