Cudmore was born at Paringa, South Australia on the River Murray. Educated at St. Peter's College and the University of Adelaide, he graduated in surgery and medicine in 1894. Five years' post-graduate study in Britain followed — for a while he was a house surgeon at London Hospital — after which he became a Fellow of the Royal College of Surgeons. Later he became lecturer in clinical surgery at the University of Adelaide and consulting surgeon at the Royal Adelaide Hospital. Cudmore had a distinguished record in World War I. He left Australia with the rank of lieutenant-colonel as consulting surgeon to the 3rd Australian General Hospital. After being invalided home with typhoid in 1916, he became consulting surgeon for the 4th Military District at Keswick. In August, 1918 he again went overseas, this time to the AIF in France, where he served for almost 12 months. After the war he resumed his post as consulting surgeon on the Australian Army Medical Corps Reserve, and in World War II was chief surgeon at No. 7 AGH, Keswick. Cudmore, assisted by Herbert Gill-Williams, started the Dental School. Elected a member of the University Council in 1927, he was Dean of the Faculty of Dentistry and president of the Dental Board. He was also a member of the advisory committee of the Royal Adelaide Hospital, where he was honorary surgeon from 1904 to 1925. Executive Council appointed him president of the Medical Board of SA in 1938.
Other interests
Cudmore was a keen motorist, and helped to found the Automobile Club of SA, of which he was elected president in 1935. He was a member of the Royal Adelaide Golf Club, and its president from 1925 to 1927.
Sir Arthur was the second son of pastoralist James Francis Cudmore, a grandson of Daniel Cudmore and cousin of Sir Collier Robert Cudmore. He married Kathleen Mary Cavenagh–Mainwaring on 13 August 1901. She survived him by a little over a week. Their children were:
Kathleen Cavenagh Cudmore married Geoffrey Champion de Crespigny on 10 June 1933. Richard Geoffrey Champion de Crespigny was a son of C. T. C. de Crespigny. A widow, she married again, to George William Symes on 30 March 1967.
