[1][2] Williams was born 18 April 1913 in Fulham, London, England, the only child of a Welsh coal miner.
[1][2] In 1965, Williams' former Southampton colleague Godfrey Lance invited him to visit the Commonwealth Scientific and Industrial Research Organisation (CSIRO) in Canberra, Australia.
[1][2] Williams' early work was on plant physiology, and more specifically on leaf expansion and the function of the Stomata.
[2][4] Recognizing the general applicability of these methods, Williams went on at CSIRO to apply them to problems other than plant taxonomy, as well as to the classification of angiosperms and grasses.
In this work, he viewed numerical taxonomy as a form of exploratory data analysis, intended more to generate new hypotheses than to confirm them.
[2][5] Williams was an avid composer of mathematical puzzles; for instance, he wrote the cross-figure puzzle "The Little Pigley Farm" (also known as Dog's Mead, Little Pigley, Little Piggly Farm, Little Pigsby, Pilgrims' Plot, or Dog Days) which he first published in 1935 in The Strand Magazine.