According to Paul McDonald at the University of Wolverhampton, it’s this 3900-year-old one, from 1900BC in Sumeria:
Something which has never occurred since time immemorial: a young woman did not fart in her husband’s lap.
(I don’t get it either!)
The oldest British joke dates back to the 10th Century and reveals the bawdy face of the Anglo-Saxons — “What hangs at a man’s thigh and wants to poke the hole that it’s often poked before? Answer: A key.”