Eskydun is exactly what Allingham wrote, and since he was based in south Donegal, Asketon, while an inspired guess from the sound, is not very likely.

His song is an adaptation of "The Lass among the Heather" written by a County Antrim poet, Hugh McWilliams, and published by him in his "Poems and songs on different subjects" printed in Belfast in 1831. Below is a series of references. Some of the variants may contain a hint as to why Allingham was impelled to use Eskydun.

The lass among the heather (1831: 106) (Roud 2894):
Ballad sheets
?Belfast ?Moore (Birmingham Univ Library, Hume Collection 310 (slightly varied 102)) [Both as The Blooming Heather]
London, Paul (BUL, Hume 136)
Unknown ?English printer [As Blooming Heather] (2806 c14(60)),
?Newcastle upon Tyne, ?Fordyce 125 [As The Blooming Heather](Bodleian 2806 c15(240)), same printer, 225 [As The Blooming Heather] (Bodleian: 2806 c13(2)),
?Scottish, Printer Unknown, [As The Blooming Heather] (Bodleian Harding B11(331))
Glasgow Poet's Box, 26 Nov. 1859 Blooming Heather (Mitchell Library, Glasgow, Poet's Box Collection),
Also Printed in Britain by Fordyce, Newcastle; J Lindsay, Glasgow; Pitts, London; Ross, Newcastle; Glasgow Sanderson, Edinburgh; Walker, Durham; Walker, Newcastle. (Roud

Song book
Dublin, JF Nugent The National Melodist (British Library 11622df19 #62) p. 34. This is patently related to the Tommy McCabe version. (see next line)

Oral versions
Monaghan, Tommy McCabe The lass among the heather O (Morton, Folksongs sung in Ulster (1970) p. 7); Antrim/Londonderry no details (Sam Henry papers [Private file I2b); Cork, Elizabeth Cronin (Ballyvourney) The Fair at Ballyally (fragmentary) (Ó Croinín, The songs of Elizabeth Cronin: traditional singer (2000) p. 154); Aberdeenshire, The Fair at Ballinaminna (Greig, Folksong of the North-East (1963), Article 44); (Shuldham-Shaw et al., The Greig-Duncan Folksong Collection vol. 4 no. 873 (12 versions)); Jeannie Robertson, The Fair o' Balnafannon (Porter & Gower, Jeannie Robertson: emergent singer, transformative voice (1995) p. 175); USA, Maine, Carrie Grover, The Fair at Baltimoreo (Grover, A Heritage of Song (n.d.) p. 20); Tunney, Where songs do thunder: Travels in Traditional Song (1991) p. 31

Literary version
Adapted by William Allingham – (Graves, The Irish Song Book (1894) p. 148)

It seems likely that the earliest print is the one of which there are two (one cropped of its cut) in the Hume collection which, I surmise, were printed by James Moore, Belfast. It is thus later than 1843 and sooner than 1849 when the Hume collection has made. The dissemination of this, in Scotland, across northern England and in London indicates how widely attractive it was. Its oral dissemination is even more remarkable. Twelve variants were collected in the Parish of New Deer in Aberdeenshire in the early 20th century.

I've got more but will wait until I'm asked!