Williamson was born in South Shields, the son of a physician. In 1870 he came to Ventnor in the Isle of Wight and served for three years in the Royal National Hospital for Diseases of the Chest at St. Lawrence. He gained further experience at the Marylebone Infirmary in London before returning to Ventnor in 1876 to take up private practice. He was so successful that he soon moved into a large house called 'Southcliff' in Belgrave Road, where he practised until his death in 1901.
Dr Williamson published Ventnor and the Undercliff in Chronic Pulmonary Diseases(1884), displaying his expertise in assessing lung disease. Something of a scholar too, his The Life and Times of St. Boniface was published posthumously in 1904. Williamson appears to have shared the saint's kindly but firm character.
Karl Marx visited the Isle of Wight in later life to recuperate and for treatment of his persistent pulmonary ailments. In 1882, Williamson ('a nice young fellow, nothing priestly about him') became the sage's physician and Marx jocularly refers to the doctor's authoritarian manner and being placed under 'house arrest'. The two seem to have got on very well and the young practitioner managed to sustain Marx's courage and optimism. The doctor's wife asked for a photograph of Marx, which he sent, with a friendly dedication. Marx left Ventnor to die in London and a note of 13.1.1883 to Williamson, explaining his departure, may be the last letter of his life.