Broadbent was born at Lindley, near Huddersfield, in 1835, the son of a manufacturer. He was originally intended for the woollen trade, but decided instead to pursue a medical career. He was apprenticed to a surgeon in Manchester, and attended Owens College and Manchester Royal School of Medicine. Broadbent was a successful student, winning gold medals in anatomy, physiology and chemistry at the first MB London exams in 1856. He qualified LSA and MRCS in 1857. He failed to be appointed house surgeon at MRI in the same year, and moved to Paris where he studied with many of the leading medical figures. He took his final MB London in 1858. Shortly after, he was appointed as obstetric officer at St Mary's Hospital, London. The rest of his career was spent in the capital; he was physician at London Fever Hospital from 1860-1879, and lectured at St Mary's. He remained at the latter hospital until 1896.
Broadbent had a reputation as one of the best clinical teachers in London. He had interests in field of cancer, but particularly in the study of neurology. He advanced 'Broadbent's hypothesis ' which explained the immunity from paralysis of bilaterally associated muscles in hemoplegia. He also made important contributions to the study of aphasia, and was an expert on heart disease and typhoid fever.
Broadbent was Censor of the Royal College of Physicians in 1888-9, and senior censor in 1895. In 1887 he delivered Croonian lecture and Lumleian lectures in 1891. He was also active in public health movements, in 1898 he was chairman of the organising committee for the National Association for the Prevention of Consumption. He was also chairman of the organising council of the British Congress on Tuberculosis which met in 1901.
Broadbent was a society doctor, whose patients included members of the royal family, including the Duke of York [George V] and the Duke of Clarence. In 1892 he was appointed physician to the Prince of Wales [Edward VII] and in 1896 physician extraordinary to Queen Victoria. He was made a baronet in 1893, and was elected fellow of the Royal Society in 1897.