Wedgwood was educated at Warrington Academy and Edinburgh University. Wedgwood had an interest in botany and horticulture, particularly the cultivation of tropical fruit and other exotic plants. He was a founder of the Royal Horticultural Society having suggested the idea in a letter to William Forsyth, head gardener to King George III, written 29 June 1801 and chairing the first meeting of the society on 7 March 1804. He lived in Abergavenny, Wales, and at Kingscote, Gloucestershire.
Wedgwood married Louisa Jane Allen (younger sister of his brother Josiah Wedgwood II's wife, Elizabeth "Bessie" Allen) in 1794. He died in Tenby, Pembrokeshire, in 1844, having had six children, one of whom died young: