The son of a builder, Cort was born in Lancaster. Although his date of birth has traditionally been given as 1740, evidence from the time of his first marriage suggests he was born in April or May 1741. As a young man, he worked as a Royal Navy pay agent in London, where his interest in the poor quality of English iron against overseas supplies developed, leading to his inventions that greatly increased the quality of iron production. He left his job as an agent in 1775. His 1768 marriage to Elizabeth Heysham had connected him to the Attwick family of Gosport, and to Elizabeth's uncle, William Attwick. Cort joined Attwick in the family iron business, living and working in Gosport.

With an increase in demand for iron goods from the Royal Navy, the ironworking enterprise in Middle Street, Gosport flourished, but was strained by a 1779 contract for re-rolling barrel hoops for the Navy Vitualling Board. To cope with this, Cort set up an iron works in Fontley, Hampshire. This was a rolling mill, and here he developed his ideas, leading to patents in 1783 for the grooved rolling mill and 1784 for his balling or puddling furnace, allowing the manufacture of crude, standardized shapes. His work built on the existing ideas of the Cranege brothers and their reverberatory furnace (where the heat is applied from above, rather than forced air from below) and Peter Onions' puddling process where the iron is stirred to separate out impurities and extract the higher quality wrought iron. The furnace effectively lowers the carbon content of the cast iron charge by oxidation. The "puddler" extracts a mass of iron from the furnace using a rabbling bar. The extracted ball of metal is then processed into a shingle by a shingling hammer, after which it is rolled in a rolling mill.

His partnership in the Fontley Mill was with Samuel Jellicoe (son of Adam Jellicoe, who had important Navy connections, and great-grand father of the hero of Jutland Admiral Jellicoe (1859-1935)). The partnership eventually turned out very badly, but initially the elder Jellicoe financed Cort's experimentation with large sums and a wharf at Gosport was purchased. Adam Jellicoe's death (1789) revealed Adam to be effectively bankrupt, and so Cort's debts to Jellicoe were called in to settle the estate. These included the patents which were thus taken over by the Government. The younger Jellicoe was given sole control of the mill and wharf although he (as Cort's partner) also had to repay his father's debts- mostly to the Navy. Cort's process was not suitable for coke-smelted pig iron and various improvements were made which did not infringe the patents. Although Cort was soon discharged from bankruptcy, his career was ruined.

Cort married twice. A short-lived marriage to Elizabeth Brown was succeeded by his marriage in 1768 to Elizabeth Heysham, by whom he had a large family. However, his business ventures did not bring him wealth, even though vast numbers of the puddling furnaces that he developed were eventually used (reportedly 8,200 by 1820) as they used a modified version of his process. He was later awarded a government pension, but died a ruined man, and was buried in Hampstead churchyard in London.

His son, Richard Cort, was cashier of the British Iron Company for a short period in 1825-6 and subsequently wrote several pamphlets which were severely critical of the management of the company. He also attacked a number of early railway companies.