Bullock was educated at Durham University and served as organist at several cathedrals before becoming organist and Master of the Choristers at Westminster Abbey in 1928. He provided music for the coronations of King George VI in 1937 and for Queen Elizabeth in 1953 as a function of this post. He later held positions as Gardiner Professor of Music at Glasgow University and director of the Royal College of Music and was active as an adjudicator and administrator.