R.J.S. Simpson was awarded the degree Master of Arts (MA) by the University of Ediburgh in 1878. Switching to the study of medicine he next qualified as Bachelor of Medicine (MB) and Master of Surgery (CM) at the same university in 1882. For some time he worked as an assistant at the Royal Westminster Ophthalmological Hospital, London. In 1883 he joined the British Army and from 1891 to 1896 held the post of assistant surgeon at the Royal Arsenal, Woolwich, London. He became a member of the Ophthalmological Society of Great Britain and contributed a paper on "A case of abdominal aneurism" to the Army Medical Report in 1894. The next year he was promoted to major and married Jean M. Halley.
Simpson came to South Africa in 1896 as surgeon-major in the Army Medical Staff. In June 1896 he was in Cape Town, where he saw a brilliant double rainbow in the fog in front of Table Mountain, extending over the bay. He described it in "Rainbow at Cape Town, June 6th, 1896" in the Quarterly Journal of the Royal Meteorological Society (1896). Soon thereafter he moved to Pietermaritzburg, where he was associated with Station Hospital. By 1897 he was a member of the South African Medical Association and served on the council of its Pietermaritzburg branch. In 1898 his address was given as Fort Napier, Pietermaritzburg. During the Anglo-Boer War (1899-1902) he served as staff officer to the principal medical officer of the South African Field Forces and was awarded the Queen's medal (3 clasps) and King's medal (2 clasps). Years later he wrote "The medical history of the war in South Africa, an epidemiological essay", which was published in the Journal of the Royal Army Medical Corps and reprinted as a monograph (London, 1911, 236p).
In 1903 Simpson was promoted to lieutenant-colonel and from 1905 to 1909 held an appointment as professor of tropical medicine in the Royal Army Medical Corps. Following his promotion to colonel in 1912 he retired the next year. However, in August 1914, after the outbreak of World War I, he was recalled to service as assistant director of medical services for the Woolwich district.