Wedgwood was the only son of Godfrey Wedgwood and his first wife Mary Jane Jackson Hawkshaw, (daughter of the great civil engineer Sir John Hawkshaw, and the poet Ann Hawkshaw) who died shortly after he was born. He was the great-great-grandson of the potter Josiah Wedgwood.
He was the first Mayor of Stoke-on-Trent. Phoebe Sylvia Wedgwood (1893–1972) remained unmarried. Doris Audrey Wedgwood (1894–1968) married Thomas Geoffrey Rowland Makeig-Jones in 1928.
Wedgwood was commissioned a Lieutenant in the 4th (Militia) Battalion, North Staffordshire Regiment in 1883.
He served as a Major in the Second Boer War, returning home in June 1902. He was awarded the Distinguished Service Order (Defence Science Organisation) in 1902 for his services during the war.
He was the first Mayor of the Federated County Borough of Stoke-on-Trent in 1910 and 1911. On the outbreak of World War I in 1914 he raised the 7th Battalion, North Staffordshire Regiment.
He was killed at the Louisiana Boiselle during the Battle of the Somme in July 1916.
His body was found and identified by James Leather, a 21-year-old bandsman and stretcher bearer. He is buried at the Bapaume Post Military Cemetery in Albert, Somme.
