John Baldwin Smithson Greathead was the son of J.H. Greathead, a member of the Legislative Council of the Cape Colony, and the next elder brother of the surveyor Walter H. Greathead*. He resided in England from 1859 to 1863 and started his schooling there. On his return he continued his studies at St Andrew's College, Grahamstown, and then spent a brief period in training at the Standard Bank. In 1873 he proceeded to the University of Edinburgh, where he obtained the degrees Bachelor of Medicine (MB) and Master of Surgery (CM) in 1879. He was admitted as a member of the Royal College of Surgeons of England the same year after further training at St Bartholomew's Hospital in London. He enjoyed a reputation for brilliant general surgery and received glowing testimonials from his teachers and chiefs. Returning to the Cape Colony he was licensed to practice in September 1879 and settled in Grahamstown to do so for the next 29 years. In 1880 he married Esther Louisa Merriman, daughter of Bishop Nathaniel Merriman and sister of the statesman John X. Merriman.
Greathead soon became highly respected as a surgeon and general practitioner and in some respects was a successor to Dr W.G. Atherstone* in the affairs of Albany. For example, he took over Atherstone's position as chief surgeon at Albany Hospital in 1880. He was also appointed district surgeon and public vaccinator of Albany, railway medical officer for Grahamstown, a member of the important Leprosy Commission of the Cape Colony in 1894-1895, and after the Anglo-Boer War (1899-1902) was a government nominee on the Colonial Medical Committee. In 1887 he was a delegate to the International Medical Congress in Washington, United States.
In January 1884 Greathead contributed a paper, "Radical cure of femoral hernia" to the first issue of the South African Medical Journal (1884-1888). Four years later he wrote on "Hysterectomy and menstruation" in the British Medical Journal. In addition to his renown as a surgeon he became well known as an ophthalmologist and travelled all over the Eastern Cape as a consultant in this field.
Greathead was a member of the British Medical Association. In February 1885 a branch of the first South African Medical Association was formed in Grahamstown, of which he was the honorary secretary. However, in March 1886 the members decided to secede and form the independent Eastern Medical Association of South Africa. In 1893 this association was replaced by the Grahamstown and Eastern Province Branch of the British Medical Association, of which Greathead was president for some time and before which he delivered many papers. Furthermore, in 1897 he was a member of the provisional committee set up to establish the (second) South African Medical Association.
He had a strong interest also in natural history, sport and photography and collected ethnological specimens. One of his hobbies was big game hunting and collecting trophies, and in its pursuit he travelled widely in southern Africa, visiting Lesotho, Zimbabwe, Zambia, Mozambique, the northern Kalahari, and East Africa. He compiled a fine photographic record and kept diaries of these trips, and contributed specimens to various museums, including the National Museum of the United States. In 1899 he presented a forest shrew to the Albany Museum, as well as a sample of gold-bearing quartz from Dobie Mine, near present Mberengwa, Zimbabwe. He donated some further mammals in 1906. When the Literary, Scientific and Medical Society of Grahamstown was revived in July 1892, mainly to manage and act as custodian of the Albany Museum, Greathead was elected as joint vice-president of the museum committee. He served in this capacity for many years and succeeded Dr H. Becker* as president of the committee in February 1905. He was a foundation member of the revived Albany Natural History Society in 1890 and of the South African Association for the Advancement of Science in 1902, and joined the South African Ornithologists' Union in 1905.
Greathead's other activities included serving as chairman of the council of St Andrew's College from 1887 to 1908. In the latter year he abandoned his practice and went farming at Norval's Pont, near his birthplace Aliwal North. However, he died from sleeping sickness only two years later while on a trip to East Africa. The director of the Albany Museum, J. Hewitt*, described him as one of the museum's most generous donors, who presented the institution with many of its larger mammals.