The Lass of Richmond Hill was written by Leonard MacNally, and set to music by James Hook; it was first published in 1789, and initially popularised by the famous singer Charles Incledon, who gave it its first public performance at Vauxhall Gardens. MacNally was an Irish barrister and playwright, and evidently wrote his verses in praise of Frances I'Anson, who he married in 1787. It was the Richmond in Yorkshire, incidentally, not the one in London: the I'Ansons lived there, at Hill House. Hook, a very prolific composer of popular songs and music for the stage, was originally from Norwich.

It was an extremely well-known song for a very long time, but rather earlier than the Music Hall, and not Scottish either, so this thread is a rather odd place to ask about it. Never mind. There's a brief discussion of it from a few years back at LYR REQ:Lass of Richmond Hill. You can see a number of broadside editions at  Bodleian Library Broadside Ballads:

The Lass of Richmond hill

It was also sometimes printed as Sweet Lass of Richmond hill, but that is not the usual title. You can see an American edition arranged for voice and piano, with words and music strangely credited to Ada Burnett (a popular stage performer of the late 19th century) at  Music for the Nation: American Sheet Music:

Sweet lass of Richmond Hill.