Probably isn't an Irish song, rather than a song that might been adopted there. That's the case more often than not.
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The Great War was what it was called at the time and after. World War I caught on after the next one, in England as well, but in more recent years, and in the context of the centenary, the older term has seemed to come back into vogue.

I always prefer "The Great War" - somehow it seems to show more respect.

I wonder if future generations might merge them together as one war, the way we do with the Napoleonic Wars (and the earlier Revolutionary War) setting aside the fact that at one point it stopped, only to start again, after a shorter interval than in the 20th century, since the peace only lasted from 1803 to 1805. It seems right in many ways actually to recognise the two 20th century wars as essentially one long conflict in two halfs, like a nightmare game of football.