It occurred to me that the Hillsborough in the first recorded instance of this name probably wasn't the site of the stadium disaster (northern England), but instead was Hillsborough Castle in County Down, Ireland. The Marquess of Downshire was made Earl of Hillsborough in 1751, which is consistent with the castle (really a Georgian mansion) being a venue for fashionable dances by 1750 when the tune was published.

I don't remember Bride's book (read through it a few years ago) but it was common for English tunebooks of the period to include a few Irish tunes. They are the main period source for Irish repertoire that old - Irish dancing masters would have used English books (or books simultaneously published in London and Dublin) for a lot of their Irish repertoire, particularly the new and trendy stuff.

I don't believe anybody used most of O'Neill's Gaelic titles in real life - he must have got a Gaelic-speaking associate to make them up.