Wilkinson was born at Springside, Todmorden, in the West Riding of Yorkshire. His father, Henry Wilkinson, was a master house painter and decorator; his mother, Ruth, worked in a local cotton mill. One of his uncles, an organist and choirmaster, had married into a family that owned a small chemical company making Epsom and Glauber's salts for the pharmaceutical industry; this is where he first developed an interest in chemistry.
He was educated at the local council primary school and, after winning a County Scholarship in 1932, went to Todmorden Grammar School. His physics teacher there, Luke Sutcliffe, had also taught Sir John Cockcroft, who received a Nobel Prize for "splitting the atom". In 1939 he obtained a Royal Scholarship for study at Imperial College London, from where he graduated in 1941, with his PhD awarded in 1946 entitled "Some physico-chemical observations of hydrolysis in the homogeneous vapour phase".