The other thread notes that the song was printed - with its melody - as early as 1714.

So, no machine guns.

I don't know of a definitive explanation of the spelling "Mallie." Conceivably it's simply "Molly" with an "a" either to indicate that "Molly" is a diminutive of "Mary" (strange but true!) or else, probably more likely, simply that the pronunciation was closer to "ah" than to "aw."

But unless somebody misguidedly sings "Maaaallie," there's no virtually no distinction audible.

These days in America, "ah" and "aw" are often (or "ahfen") substituted for one another. "Mawlly" and "Mahlly" coexist - though the name is now rare.