In fact, there is a variant noted in Wexford, and cited in the Roud Folk Song Index, which I missed last night. It appeared in Joseph Ranson's Songs of the Wexford Coast (1948; reprinted 1975). From the opening line quoted, 'Twas in the month of January, way down in the southern seas, it doesn't seem to be the one we have here. Nevertheless, I may have been a little unfair to the Wolfe Tones this time. Their official website, incidentally, does have a little background on some of the songs they've recorded (not always terribly accurate) but they have no comment to make on this one.

There's a brief sound-sample at Amazon: Wolfe Tones: A Sense of Freedom; not long enough to identify the tune with any certainty or to supply the missing words.

A broadside example can be seen at Bodleian Library Broadsides:

The merman ("'Twas in the month of January, away in the Southern seas ...")  Printed between 1850 and 1899 by T. Pearson, 4, & 6, Chadderton Street, Oldham Road, Manchester. This too has a chorus (the DT text does not)

Blow ye winds I oh, blow ye winds y'heave ho,
Clear away the morning dew, and blow ye winds I oh!

This would suggest that the broadside text at least was intended to be sung to some form of Clear [Blow] Away the Morning Dew; the sound clip above does indeed resemble that a bit, particularly in the snatch of chorus. The sea song Blow Ye Winds [in the Morning] is also related somewhere along the line. There's no evidence that the Vermont set was sung to any of the tune variants associated with that lot, of course, though it may have been.

The Merman itself has the look of a 19th century comic song originating on the stage; it doesn't seem ever to have been widespread in tradition.