There are two known good possibilities, "Tweedside" and "You London lads be sorry/merry". There's no evidence to go any further. I suspect that "Wallifou fa' the cat" consists of verses from two different songs. The opening verse could be to either of the two tunes mentioned. What tune might be the proper for the rest, I have no idea. Ask Murray on Saltspring. He's the one that got it from Herd's 'Scots Songs' and contributed it to DT, and he knows a lot about Scots music, as well as songs.