Adamson was born in Shildon, County Durham, and was the nephew of civil engineer Daniel Adamson, a founder and first chairman of the Manchester Ship Canal Company. Joseph Adamson’s father was an engine driver and later a coal merchant in Doncaster in Yorkshire. In 1858 Adamson was apprenticed at his uncle’s business, Newton Heath Boiler Works outside Manchester, where he became foreman in 1863, managed the works and technical drawing department from 1865, and from 1867 to 1873 was responsible for commercial and technical management. During this time Adamson gained a good deal of experience in the development and improvement of boilers and their constituent components and materials.

In September 1874 Adamson went into partnership with Henry Booth, establishing the boilermakers Joseph Adamson & Co. in Hyde, near Manchester. When Booth retired in 1887, but Adamson continued to run the firm. The firm expanded and in particular prospered thanks to innovations in hydraulic plate flanging for the general boiler trade and the development of electric travelling cranes.

In 1902 Adamson's sons Daniel and Harold became their father's business partners.

Adamson became a director of several companies, including the North Lincolnshire Iron Company. He was a member of the Institute of Mechanical Engineers, the Iron and Steel Institute, and of the Institute of Metals. In 1908 Adamson became President of the Manchester Association of Engineers.

Adamson died at his home, Oaklands, Hyde, Cheshire, aged 77, after a brief illness.