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.
Cudmore was a keen motorist, and helped to found the Automobile Club of SA (later Royal Automobile Association), 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.
Cudmore was appointed a Companion of the Order of St Michael and St George (CMG) on 1 January 1936.
He married Kathleen Mary Cavenagh-Mainwaring (11 February 1874 – 8 March 1951) on 13 August 1901. (Kathleen's father, Wentworth Cavenagh-Mainwaring had represented the district of Yatala in the South Australian House of Assembly from 1863 to 1881 and served as Commissioner of Crown Lands and Public Works.) She survived him by a little over a week. Their children were:
- Rosemary married her cousin Rafe Gordon Dutton Cavenagh-Mainwaring, of Whitmore Hall, Whitmore, Staffordshire on 11 June 1931
- Kathleen married Geoffrey Champion de Crespigny on 10 June 1933. A widow, she married again, to George William Symes (1896–1980) on 30 March 1967.
- "Government Gazette". The Advertiser. Adelaide: National Library of Australia. 18 January 1895. p. 7. Retrieved 15 May 2013.
- "Death Of Arthur Cudmore". The Advertiser. Adelaide: National Library of Australia. 1 March 1951. p. 2. Retrieved 14 May 2013.
- "Dr. Cudmore and the Adelaide Hospital". The Register. Adelaide: National Library of Australia. 17 July 1925. p. 8. Retrieved 15 May 2013.
- It's an Honour. Retrieved 29 August 2015
- "Death of a South Australian Colonist". The Argus. Melbourne: National Library of Australia. 10 January 1895. p. 5. Retrieved 15 May 2013.
- "Family Notices". The Advertiser. Adelaide: National Library of Australia. 5 July 1933. p. 14. Retrieved 15 May 2013.
- Roger André, 'Symes, George William (1896–1980)', Australian Dictionary of Biography, National Centre of Biography, Australian National University, accessed 15 May 2013