The son of Francis Wedgwood and his wife Frances Mosley. He was a great-grandson of the potter Josiah Wedgwood. He married Emily Catherine Rendel, daughter of the engineer James Meadows Rendel, on 6 November 1866, and they had five sons, one of whom died in infancy:
- Francis Hamilton Wedgwood, JP, High Sheriff, (1867–1930)
- Clement Henry Wedgwood (1870–1871) died in infancy
- Josiah Clement Wedgwood, 1st Baron Wedgwood (1872–1943)
- Sir Ralph Wedgwood, 1st Baronet (1874–1956) 1st baronet, chief officer of the LNER for 16 years.
- Arthur Felix Wedgwood (1877–1917), killed during the First World War 
- Josiah C. Wedgwood, Wedgwood Pedigrees, 1925