Alfred Moffat and Frank Kidson (Irish Minstrelsy, 1897, p.21) had this to say:

"As Rakes of Mallow, this air occurs in Burk Thumoth's Twelve English and twelve Irish airs, London, circa 1745-50, and as The Rakes of London in Johnson's Two Hundred Country Dances, vol. vi., London, 1751. In the latter publisher's Compleat Tutor for the Guittar, circa 1755, it is styled Rakes of Marlow, and in Aird's Selection of Scotch, English, Irish and Foreign Airs, vol. ii., Glasgow, 1782, The Rakes of Mall. Arnold has made good use of the tune in his opera, Auld Robin Gray, 1784. The "Rakes" were the young gentlemen of the last century who frequented the "waters" of Mallow."

Mallow (Co. Cork) was a well-known spa town at that time. Moffat prints a song of uncertain attribution which is rather racier than the one I too remember from primary school; the words of the latter were apparently written by one A. H. Body, but I don't know who he might have been. So; the balance of probability would suggest an Irish origin for the tune, but it has also been current in England (and, of course, America) for some 250 years, and is used in Northwest Morris among other things.

There is certainly no harm in mixing English and Irish tunes; we have, after all, been doing exactly that, in both countries, for hundreds of years. Brighton Camp (another tune of disputed origins!) and Rakes of Mallow are often played together in sessions. More information at  The Fiddler's Companion:

Rakes of Mallow