Fleming was the second son of Colonel J. W. F. Sandwith and was born at Belgaum in India. He was educated at Charterhouse School and then studied Medicine at St Thomas’s Hospital, qualifying in 1876 (M.R.C.S). He gained his M.D. from the College of Medicine in Newcastle upon Tyne in 1893.
Fleming was an ambulance surgeon in the Turco-Serbian war of 1876, and in the Russo-Turkish campaign in 1877-8; he was present at the fighting at Shipka Pass, and served on Baker Pasha's staff during his retreat across the Rhodope Mountains. In 1883 he went to Egypt to combat a cholera epidemic, and acted as vice-director of the Public Health Department of the Egyptian Government until 1885. He later served as Professor of Medicine in the Egyptian Government Medical School, and physician to the Kasr-el Ainy Hospital, Cairo. He was appointed a Fellow of the Royal College of Physicians in 1900, and in the same year became senior physician to the Imperial Yeomanry Hospital at Pretoria, and served throughout the South African war. He appears to have returned briefly to Egypt, but returned to Britain in 1904 to lecture on tropical diseases at St. Thomas's Hospital and the London School of Tropical Medicine. In January 1913 he was appointed Professor of Tropical Medicine at the London School of Tropical Medicine. In 1915 he was elected to serve as president of the Durham Medical Graduates Association for the next year and to represent the association on the court of Governors for the college (Medical Gazette Volume XVI, pp. 13-14).
Fleming died of natural causes in his sleep, however the strain of his wartime service is stated to have contributed to his death.
“we would add a few words about "the man" rather than "the doctor". For those who really knew him there was a charm about his personality which greatly endeared him to his friends. Very quiet, almost reserved in manner, with a half-cynical pose which was more assumed than real and which was relieved by a most delightful sense of humour, he was one of the kindest of men, always ready to help, saying little or nothing about it, and never grudging trouble in doing it. He had a wonderful power of sympathy and looking at things from the point of view of those who sought his advice - a power which greatly added to his usefulness to the world.
His health latterly had not been good, and some time ago he underwent a severe operation, from which however he made a good recovery. But he felt the strain of the last two years in Egypt, and the heat, which was more than usually intense, greatly taxed his strength. When he returned home about two months ago his friends could not fail to recognise that he was worn and jaded. He was still anxious to work, and his retirement from the army was a great disappointment to him; but he took it quietly with a smile, and in the spirit of a soldier. He died gently in his sleep - a happy ending to a strenuous life. British Medical Journal 2nd March 1918”