Percy Scholes' book on the song gives more detail. If you're really curious that's the place to look. The attribution to John Bull looks pretty convincing to me.

it has no lilt or lift that I would normally associate with Irish melodies.

Phooey. The Irish simply didn't have any galliards.

And it is hardly a march - it's too slow.

At galliard tempo it's a bit too fast to be a march. Compare it with a 3/4 Highland pipe march like "Lochanside".