Sir Weldon, who assumed his mother's maiden name as an additional surname in 1924, qualified M.B., B.Ch. from Oxford and trained at St. Bartholomew's. After serving as Assistant Medical Officer of Health at Willesden in North London,he joined the Ministry of Health in 1927 and was Deputy Chief Medical Officer of Health from 1940 to his retirement in 1956. Among the subjects upon which Sir Weldon reported to the Ministry were undulant fever (brucellosis) and tuberculosis of bovine origin.
From 1937 to 1939, he was a member of the Interdepartmental Committee on the Nursing Services, whose interim report (G.6) laid the foundations for all subsesquent improvements in nursing. Work on a final report was curtailed by the outbreak of the Second World War, as were Sir Weldon's investigations into the use of snake venom for pain relief and the treatment of epilepsy.
He was President of the Royal Society of Medicine's Comparative Medicine Section 1954/55. He also undertook work for the international medical community, reporting on bovine tuberculosis to the International Bureau of Public Hygiene (F.12) and representing New Zealand on the Bureau's Permanent Committee (this body was later absorbed into the World Health Organisation). He was President of the Joint Food and Agriculture Organisation / World Health Organisation Committee on Brucellosis, and did much to further the cause of veterinary education by his active support of the Royal Veterinary College and the Veterinary Educational Trust.
After his retirement, Sir Weldon became President of the Haemophilia Society. After his move to Oxford in 1964 he was closely involved in the work of the Oxford Haemophilia Centre.
He died in 1980.