I think "medieval" is pushing it.

The "Tom a Lin" name began to be replaced by "Brian O'Linn" in the eighteenth century. (See discussion on other threads, above.)

The well-known form of the song, with "no breeches to wear," etc., seems to come from nineteenth-century broadsides. To judge from printed references, it seems to have been exceptionally popular in Britain, Ireland, and America.

The popular Irish jig tune called "Brian O'Linn" seems never to have been used for the song; perhaps the jig was elaborated from something simpler, or perhaps it was once associated with some stage routine about B O'L.

W. H. Maxwell's comic novel, "Brian O'Linn; Or, Luck is Everything" appeared in 1848 and went through many editions. The American Samuel D. Johnson's one-act play of "Brian O'Linn" appeared soon after.

Bawdy versions also exist, and one is said to have been recorded in Stan Hugill's lost manuscript collection of chanteys.

So if anybody knows one, they should post it.