Wanklyn's father was born in Argentina and brought up in Ayrshire, Scotland. His ancestors had moved to Argentina in the 19th century. His maternal grandfather was managing director of the Mercantile Bank of the River Plate until its collapse in 1875. His grandfather, Frederick, died soon afterwards and his grandmother Elizabeth discovered his estate had been mismanaged. Left destitute, she headed back to England with her eight children including William. Ill-fortune struck again during their return when the vessel in which they were travelling, the Royal Mail Steam Packet Company, "Boyne", ran aground on Friday 13 August on the rocks off the Isle of Molene, near Brest, France. The family lost most of their belongings.