The set recorded by Barrand and Roberts was taken from Stephen Sedley's book The Seeds of Love (1967), with an additional verse tacked on at the end.  Though popular 30 years ago, Sedley's material isn't much used, now, as most of the songs were "mix and match" collations from all manner of disparate sources, with hardly a single genuinely traditional song among them.  Reynardine, for example, was described as "Text collated from two 19th-century broadsides (The Mountains High) by Such and Pitts, plus sets collected in Ireland by Joyce and Hughes and the version sung by A.L. Lloyd, which he had originally from Tom Cook, of Eastbridge, Suffolk.  Tune collected by Merrick from Henry Hills, a Sussex farmer."

A bit of a Dog's Breakfast, in short.  The information about Lloyd's source is useful, and I'm grateful to Animaterra and Becky for making me think of checking that reference.  I may as well tie a few loose ends, here; the Such and Pitts broadsides can be seen at  Bodleian Library Broadsides:

The mountains high  Printed between 1849 and 1862 by H. Such, 123, Union Street, Boro' S.E., London.

The mountains high  Printed between 1819 and 1844 by J. Pitts, Wholesale Toy and Marble Warehouse, 6, Gt. St. Andrew Street, Seven Dials, London.

The final verse added by Roberts and Barrand is found in both the above.  The tune mentioned was first published in the Journal of the Folk Song Society (vol. I, number 3, 1904).  W. Percy Merrick got it from Henry Hills (c. 1831-1901), of Lodsworth, near Petworth in Sussex.  He had learned it from his mother, and had only a partial text, as follows:

One night upon my rambles an Irish girl I spied;
Your beauty so enticed me, I could not pass you by.
So it's with my gun I'll guard you all on the mountains high
So it's with my gun I'll guard you all on the mountains high.

She said, "Kind sir, be civil, my company forsake,
For in my own opinion I think you are some rake,
For if my parents they should know, my life they would destroy
For a-keeping of you company, all on the mountains high."

The set published by Herbert Hughes in Irish Country Songs (vol. I, 1909) has a tune fairly close to Lloyd's Sussex one; the text, again, is fragmentary, and described by Hughes as "Fragment of an Ulster Ballad: Donegal version".

If by chance you look for me
Perhaps you'll not me find,
For I'll be in my castle,
Enquire for Reynardine.

Sun and dark I followed him,
His eyes did brightly shine;
He took me o'er the mountains,
Did my sweet Reynardine.

As has been mentioned earlier, Hughes noted, "In the locality where I obtained this fragment Reynardine is known as the name of a faery that changes into the shape of a fox".  This is the only piece of evidence that supports the modern, widely-held assumption that this is a supernatural song; in common with some others, I rather suspect that Hughes' informant was just winding him up.

Midis of these tunes will go to the  Mudcat Midi Pages;  until then, they can be heard via the  South Riding Folk Network  site:

One Night Upon My Rambles  Henry Hills' tune, as recorded by Barrand and Roberts.

Reynardine  Donegal tune, as published by Herbert Hughes.