The Streets of Laredo is certainly sung to the same tune as the Bard of Armagh, but chances are that the tune had already been recycled. The Bard of Armagh was someone called "Bold Phelim Brady", but I think his only claim to fame (if he even existed) is the song about him.

BTW, we owe a lot to the Chieftains, but much of that album just sucks. In keeping with Paddy Moloney's less attractive character trait (and the "authorised biography" of the Chieftains), it's just one long exercise in name-dropping.