John Wilson attended school in Kendal. From there he entered Peterhouse, Cambridge and he was the Senior Wrangler in 1761. This means that he was the best of all the First Class students to graduate after taking the Mathematical Tripos.
Before he took his final examinations Wilson had already gained a strong reputation and he had also attracted considerable attention by defending Waring, who was the Lucasian Professor of Mathematics, from strong attacks which had been made on him as a result of his text Miscellanea analytica.
In 1764 Wilson was elected a Fellow of Peterhouse and he taught mathematics at Cambridge with great skill, quickly gaining an outstanding reputation for himself. However, he was not to continue in the world of university teaching, for in 1766 he began a legal career. It was a highly successful career, too. He became a member of the Court of Common Pleas in 1786 which had been set up to make judgements in cases of civil disputes between individuals. At the time that Wilson served on this body it was one of three courts which dealt with common-law business.
He is best known for Wilson's theorem which states that
... if p is prime then 1 + (p - 1)! is divisible by p
This result was first published by Waring, without proof, and attributed to Wilson. Leibniz appears to have known the result. It was first proved by Lagrange in 1773 who showed that the converse is true, namely
... if n divides 1 + (n - 1)! then n is prime.
Almost certainly Wilson's theorem was a guess made by him, based on the evidence of made cases, which neither he nor Waring knew how to prove.