Bush was born in London and studied at the Royal Academy of Music there. Later he studied musicology and philosophy in Berlin and later still had lessons with the composer John Ireland. He studied the piano under Benno Moiseiwitsch and Artur Schnabel. From 1925 to 1978 he taught at the Royal Academy of Music.
He was known as an outspoken advocate of Marxism, holding posts as conductor of the London Labour Choral Union and, from 1936, chairman of the Workers' Music Association. This influence can also be seen in many of his works, including the operas Wat Tyler (1948-50) and Men of Blackmoor (1954-55), and his Piano Concerto which has a communist text declaimed by a male chorus in the last movement.