In 1794 Mellish inherited Bush Hill Park from his aunt, Catherine Mellish (née Gore), widow of Joseph Mellish.
With his brother John, he owned the business of John and William Mellish & Co. Having served as a director and Deputy Governor, Mellish was made Governor of the Bank of England from 1814 to 1816.
Mellish died at his residence on Bishopsgate Street, London [4] on 8 June 1838, aged 73, and was buried at All Saints' Church, Edmonton.[5] He never married. After his death, the proceeds of his estates were divided between his nephews and nieces.