As far I can see, nobody has mentioned that one of the earliest copies of this song is in the British Library, published circa 1850 and their CPM shows that it's a republication from 'The Book of Beauty for the Queen's Boudoir' (issued between 1843 and 1847) with music by Charles William Glover and words by Edward Mordaunt Spencer.

There's also, as the preceding post notes, the publication of the poem in Spencer's 1846 anthology (it can be viewed online thanks to the Hathi Trust Digital Library) which has two other odes to his Irish Mary (though it's odd that here the music for 'The Rose' is credited to Charles's brother, Stephen).

As already has also been noted, this song was featured in John McCormack's movie 'Song 'o the Heart'. I've a vague recollection that years ago McCormack's recording was introduced on BBC Radio 3 with a story that someone connected with the film (one of the cast?) knew an obscure Irish song from their mother, sung it to McCormack, he liked it and sung it in the film. This may be untrue but it chimes in with I and others here have found: some publications in Britain and the US in the mid-19th century, then nothing until 1930, the year the movie was released. The song would indeed seem to owe its current popularity to Count McCormack.

There seems little doubt that, based on the evidence, the poem is by E. M. Spencer. Where then does the attribution to W. P. Mulchinock and the story attached to him come from? A reference on ancestry.com admits that the poem is not in his only collection published in 1851. I suspect that when the song was becoming (newly?) popular in 1930s Ireland, its true provenance as a ballad written by two Englishmen (Spencer may have been Anglo-Irish) in a book for Queen Victoria was unacceptable so another origin was invented for it (We Irish love stories). Perhaps Mulchinock did write an ode for his Mary which is now forgotten and the story was transferred.

As for the third stanza, the India verse, if Vin Garbutt was the first to sing it, it probably comes from a London priest as he claims. It's rather reminiscent of Thomas Moore (e.g. Savourneen Deelish) so it could be a pastiche. Whether the author was deliberately trying to add credence to the Mulchinock myth is an interesting question. I hope someone will have more info on this.