After a period of research in the Cavendish Laboratory, Cambridge, in 1909 Wilton was appointed assistant lecturer in mathematics at the University of Sheffield. On 5 September 1910 at St John's Church, St Teath, Cornwall, he married Annie Martha Gladstone (d.1932). The University of Adelaide awarded him its first D.Sc. in mathematics in 1914; by 1916 he had published a prodigious twenty-five research papers on partial differential equations, hydrodynamics and elliptic functions.
During World War I he suffered as a pacifist. Before the introduction of conscription, Wilton had undertaken X-ray work in a Sheffield hospital, but in 1916 was directed to London where repetitive tasks and financial privation contributed to his breakdown in June 1918. After the war he joined the Society of Friends and taught briefly at one of their schools. Having accepted an assistant lectureship in mathematics at Owens College, Victoria University of Manchester, under (Sir) Horace Lamb, in 1919 Wilton was appointed (Sir Thomas) Elder professor of mathematics at the University of Adelaide.
Back in Adelaide in 1920, he found the mathematical courses much the same as those he had attended as an undergraduate. Although he revised the curricula, as well as those at the senior school level, he was not successful in encouraging honours students, only eight of whom graduated between 1920 and 1940. In 1926-34 Wilton published twenty-six research papers on analysis and number theory: his sustained research gained him a doctorate in science from the University of Cambridge in 1930 and the Lyle medal in 1935 from the Australian National Research Council.
In his 1954 address on Australian contributions to mathematics (Sir) Thomas Cherry rated Wilton's research as second only to that of J H Michell. H S Carslaw and G H Hardy regarded Wilton's work as that 'of a fine mathematician, with admirable taste and a natural inclination towards deep and difficult problems É He might perhaps have made a bigger name if his taste had been less fine, and he had been content to work in fields which offer cheaper rewards'.
Beyond mathematics, Wilton's prime interests were music and literature. He read Dante's Divine Comedy in Italian several times a year for twenty years. The Faculty of Arts at the University of Adelaide acknowledged him as 'a scholar of powerful and sensitive intellect who united highly specialised learning with a broad and deep culture, and a gentleman of unassuming demeanour, warm sympathy, just judgment and steadfast devotion to principle'. Active in the Society of Friends, he was clerk (1934-37) of the General Meeting for Australia, twice president of the South Australian Council of Churches, Australian treasurer of the Save the Children Fund and was associated with the Student Christian Movement.
A stroke in 1941 so impaired Wilton's speech and memory that 'he had to learn mathematics afresh, even his multiplication tables'. In 1944 he returned to lecturing, but died of a cerebral haemorrhage on 12 April at the Wakefield Street private hospital, Adelaide, and was buried in West Terrace cemetery. His wife Winifred Aimée, née Welbourn, whom he had married on 25 May 1936 at the Friends Meeting House, North Adelaide, survived him, as did their daughter.