I've mentioned this in some other thread here, but the tune is also used for a dance/song from the Gyimes/Felcsik area of eastern Transylvania, with a refrain beginning "Golya, Golya" ("Stork, stork" in Hungarian) at the point where the English has "Glory, glory" (the sounds of the words are disconcertingly similar). The dance is a round-the-room march rather like a Gay Gordons. The last few notes of the tune are a bit different from the American version.

I asked where they got the tune and the consensus was that it was one of theirs. If they're right, one guess is that it might have gone to America at the right time with emigrants fleeing the troubles of 1848, but a lot of Central European music was travelling the world around then without the impetus of insurrectionary violence driving it - polkas and waltzes, in particular.

But I can't find a transcription of the Hungarian words or even a coherent explanation of what they were about. It will be a regular item in Budapest "tanchaz" ceilidhs, so somebody who goes to those might be able to help.