Michael,
It's quite possible that that particular 'Bold Grenadier' tune is derived from the Clifton song, or even via 'Cushy Butterfield'. As very few people were noting down broadside tunes before the 1880s it's almost impossible to say.

I can see the similarity with 'Bunclody' and this is (IMO) a particularly beautiful version of the tune, but again which came first is impossible to say, unless a variant of 'Bunclody' can be found in something like Bunting or any of the Irish collections with tunes before the 1860s. Harry was rather partial to Irish songs but usually set tunes himself to Irish poetry, e'g', 'Rocky Road to Dublin', 'Lannigan's Ball', 'Darby Maguire', 'The Waterford Boys', 'The Wedding of Biddy McGrane', 'The Island of Green' etc.