Frank Kidson of Leeds (1855- 1926) was a serious collector who did not interfere with the material he found, though he was of course subject to the normal reservations of the time where it came to what they used to call "coarse" material.  Besides several books on traditional song and dance, he published studies of The Beggars Opera, the Leeds Pottery, British music publishers, and so on.  Much of his MS collection is now in the Mitchell Library in Glasgow, but recently Paul and Liz Davenport unearthed a number of his notebooks amongst material bequeathed by Anne Gilchrist to the Vaughan Williams Memorial Library at Cecil Sharp House in London.

Alfred Moffat, born in Edinburgh (1866-1950) was not really a collector, but a compiler and arranger.  He collaborated with his friend Kidson on A Garland of English Folk Songs and The Minstrelsy of England, and also published anthologies of Scottish and Irish songs arranged for voice and piano, collections of nursery rhymes, and so on.

There are two completely different songs called Irish Molly (and variations thereon).  The traditional one has been found in England, Scotland, Ireland, the USA and Canada; Martin Ryan posted a text which apparently derived from an American songbook of the early 19th century here:  Irish Molly-O;  it was set to the tune later used for The Sash My Father Wore, it seems.  Probably not possible to establish its exact origins; it looks like a song made in England or perhaps Scotland on an "Irish" topic rather than an Irish song per se, but I don't think there's any definitive answer.  It was certainly popular in the early-to-mid 19th century, and there are quite a few broadside copies at  Bodleian Library Broadside Ballads,  together with other songs naming Irish Molly as tune.  Here is one:

Irish Molly, O!  Printed between 1819 and 1844 by J. Pitts, Printer, Toy and Marble Warehouse, 6, Great St. Andrew Street, Seven Dials [London].

The other song,  MY IRISH MOLLY-O  is an American "stage Irish" comic vaudeville piece, popularised in recent years by De Dannan and, subsequently, taken up by many others.  It may perhaps have been a parody of the older song, but I haven't heard that earlier one and can't really comment.