At Beth's Notes, she uses Wikipedia as her source. The heading of the song claims it is from 1715, however there are no confirming sources I could access.
Here's the link to the Wikipedia Article about Aiken Drum. It mentions the source as "Jacobite Relics," published in 1820 by James Hogg, but I could not find a digital copy of that publication. Here are the original lyrics, as cited by Wikipedia:
Ken ye how a Whig can fight, Aikendrum, Aikendrum
Ken ye how a Whig can fight, Aikendrum
He can fight the hero bright, with his heels and armour tight
And the wind of heavenly night, Aikendrum, Aikendrum
Is not Rowley in the right, Aikendrum!
Did ye hear of Sunderland, Aikendrum, Aikendrum
Did ye hear of Sunderland, Aikendrum
That man of high command, who has sworn to clear the land
He has vanished from our strand, Aikendrum, Aikendrum,
Or the eel has ta'en the sand, Aikendrum.
Donald's running 'round and 'round, Aikendrum, Aikendrum,
Donald's running 'round and 'round, Aikendrum
But the Chief cannot be found, and the Dutchmen they are drowned
And King Jaime he is crowned, Aikendrum, Aikendrum
But the dogs will get a stound, Aikendrum.
We have heard of Whigs galore, Aikendrum, Aikendrum
We have heard of Whigs galore, Aikendrum
But we've sought the country o'er, with cannon and claymore,
And still they are before, Aikendrum, Aikendrum
We may seek forevermore, Aikendrum!
Ken ye how to gain a Whig, Aikendrum, Aikendrum
Ken ye how to gain a Whig, Aikendrum
Look Jolly, blythe and big, take his ain blest side and prig,
And the poor, worm-eaten Whig, Aikendrum, Aikendrum
For opposition's sake you will win!
In a blue box in the article, Wikipedia quotes the following passage from George Farquhar Graham 'The Songs of Scotland, Adapted to Their Appropriate Melodies, Arranged with Pianoforte Accompaniments, Volume 3, 1849, p. 27:
"AIKEN DRUM." - We have been unable to obtain any satisfactory information regarding the origin of this air. Some persons consider it as one of the most ancient of our Scottish airs. We do not. In our boyhood it used to be sung to ludicrous but unmeaning stanzas, beginning-
"There lived a man in our town, In our town, in our town, There lived a man in our town, And his name was Aiken Drum."
We were told that this man wore a strange coat, with buttons of "bawbee-baps," and that "he played upon a razor." James Hogg, in the second series of his "Jacobite Relics," page 22, gives another "Aiken Drum," which he interprets politically with the aid of Sir Walter Scott-
"Ken ye how a Whig can fight, Aikendrum, Aikendrum?" &c.
The air to which these political stanzas are set is quite different from the air here given, with words written for it by a friend of the Publishers. Hogg quotes also the first stanza of another Aiken Drum, in which that personage is said to have come from the moon. In Mr. R. Chambers' "Popular Rhymes of Scotland," there is a vigorous ballad of thirty stanzas about another Aiken Drum, called "The Brownie of Blednock," written by William Nichol-son, & Dumfriesshire peasant. Another song to this air, beginning, "A piper came to our town," will be found in the Appendix."
Photos of the publication, with links to the pages can be found below.
Triplets : Comprising, The Baby's Opera, The Baby's Bouquet, and The Baby's Own Æsop, published in 1899; has a lot of music meant for children and is the earliest version of the modern melody that I could find.
Found a mention of Aiken Drum in the "Brownie of Blednoch" poem, by William Nicholson, written in 1849