Beardmore was born in Greenwich, London, in the family home. His father, also William Beardmore, was a mechanical engineer, working for the General Steam Navigation Company in Deptford. In 1861 his family moved to Glasgow, where his father bought the Parkhead Forge, a steel mill and supplier to the thriving shipbuilding and railway industries on the Clyde in the east end of the city.
Beardmore was born at his family home in Greenwich, London, to William Beardmore, a mechanical engineer working for the General Steam Navigation Company in Deptford. In 1861, the family relocated to Glasgow, where his father acquired the Parkhead Forge, a steel mill serving the shipbuilding and railway industries on the Clyde.