A short form of the song was well known in the 1940s via a commercial recording by Barbara Mullen which was frequently played on the radio; that would be the form of it that I remember from childhood in the 1950s, I expect; unless it was the later recording by Shirley Abicair. It was described as "a traditional Irish song", and no doubt it was in that form, though similar forms of words were current in oral tradition in the South of England at the beginning of the 20th century.

He may go or he may stay, he may sink or he may swim,
I do think in my own heart I am quite as good as him.
Aye and if he get another girl we both will agree.
I will defy the lad for ever - let him go, farewell he.

(George Smith, Fareham, Hampshire, 1906. Quoted in Reeves, The Everlasting Circle, 108-9).

It's generally considered to belong to the group of songs found in tradition as Farewell He, Fare Thee Well Cold Winter, and so on; examples of which are known from Ireland, England and Scotland; and, of course, America and Canada. How far each is related to the others isn't always easy to tell, as not all of the characteristic verses always appear, and some examples may just be coincidental assemblies of floating verses.

The "Let him go let him tarry" sequence, for instance, seems to have been introduced from elsewhere; and a couple of examples from the South of England acquired the chorus of an American song, I'll be all Smiles Tonight (1879), somewhere along the way. English sets tend to have tunes very reminiscent of the Music Hall; more so than the Irish set here, though it wouldn't surprise me at all if that one had a stage origin, as Jennie suggested.

Examples fall into two main groups in the Roud Folk Song Index: 803 (which includes the set I quoted from above) and 1034 (which includes the set of Let Him Go Let Him Tarry printed in Walton's Treasury of Irish Songs & Ballads, 1947).

There have been discussions here relating to Farewell He in the past, and others, mostly irrelevant, that have touched on Let Him Go. One thread at least contains some useful references:

Farewell He/She? (Adieu to Dark Weather)

The DT files are

Let him go, let him tarry (source not identified)
Farewell He (apparently from an unidentified Jean Redpath record; traditional source not credited)
All Around my Hat (modern collation which incorporates material from Farewell He)

Broadside editions of Farewell He can be seen at Bodleian Library Broadside Ballads. Some are transcribed in the thread indicated above, but this should provide a full list:

Farewell He