From Joyce's Notes
Ref Don Gifford (Ulysses Annotated) knows one of these songs, too:
"Kick the Pope – One version of this Orange faction chant is a nagging street rhyme, "Tooral, looral, kick the Pope; / Hang him up wi' taury rope" (Leslie Daiken, Out Goes She; Dublin Street Rhymes [Dublin, 1963], p. 20. ['Taury' = 'tarry'.]"

From childhood memories the tune that was probably used for the "childrens" song in Joyce's notes would be the tune known in the UK "The British Army" which is at Tune the British Army Words "If it wisnae for the unions"
I'm sure some visting fiddle or accordion player will be able to Identify the original? name of the tone