It's quite old as these things go, and was licensed to the publisher Thomas Orwin in 1588, though no broadside copies survive. It appeared in Thomas Ravenscroft's Deuteromelia (1609):

Freemens songs of 4 Voices.

...and in various publications during the 17th century. A different tune, with the "wha's fu' now" refrain, appeared in Henry Playford's Original Scotch Tunes (1701); this seems to be the earliest version we have from Scotland, where it also appears in a number of guises.

Ann Gilchrist (Journal of the English Folk Dance and Song Society, IV (3) 1942, 118-121) quotes examples from England and Scotland, and remarks on "...a confusion between 'fool' and 'full' (fou'=drunk) on one side or other of the Border", but draws no conclusions as to any "national" origin for the song and its relatives, pointing instead to the occurrence of similar songs and themes throughout Britain and Ireland, in most of the languages spoken here.