Born in London, Hess studied at the Royal Academy of Music under Tobias Matthay. Her debut came in 1907 when she played Ludwig van Beethoven's Piano Concerto No. 4 with Thomas Beecham conducting. She went on to tour through Britain and North America.
She gained even greater fame during World War II when, with all concert halls closed, she organised a series of lunchtime concerts at the National Gallery, playing in many herself. For these concerts, she was created Dame in 1941.
Hess was best known in Wolfgang Amadeus Mozart, Ludwig van Beethoven and Robert Schumann, but had a wide repertoire ranging from Domenico Scarlatti to contemporary works - she gave the premiere of both the Piano Sonata and Piano Concerto by Howard Ferguson. She also played a good amount of chamber music, and performed in a piano duo with her cousin Irene Scharrer. She made a well known arrangement for piano of the chorale prelude from Johann Sebastian Bach's Cantata No. 147 (Herz und Mund und Tat und Leben) under the title "Jesu, Joy of Man's Desiring".