Bury was born at Salford, and after studying at Owens College, moved to University College London in 1872, where he qualified. He held appointments at MRI and Manchester Children's Hospital before specialising in neurology. He joined the honorary staff of MRI in 1889, and became professor of clinical medicine at the University of Manchester in 1911.
Bury's area of specialism was neurology and he published widely on nervous diseases, including peripheral neuritis. He published Diseases of the nervous system in 1912, and he also published Clinical medicine (1894) which ran to several editions. He was president of both the Manchester Medical Society and the Pathological Society of Manchester.