Jack- The "Irish" one was published in the us, and creditied to Harrigan (of Harrigan and Hart). I'm trying to find a date for the American one--Bascomb Lamar Lunceford claims it, but I'm not sure if it was written by him or just published. Little Brown Jug is mid-1800s; most of the songs mentioned above are 20th century. (Copper Kettle is even the latter half of the 20th century, as is Chig-a-Lug).
I did find that the first use of "mountain dew" for moonshine liquor is 1829, as per Merriam Webster. Just remembered an 1860-ish parody which starts,
Just before the battle mother,,
I was drinking Mountain Dew...