As to "Down By the Sally Garden," a couple of points:

1. You can sing many songs to the scansion of others — "Home on the Range" to "Star Spangled Banner," just to pick an example out of my cluttered head.

2. The song history of "Down by the Salley Garden" is quite interesting. Its ultimate origin is uncertain, but the Yeats connection is fairly clear. This deserves a thread of its own, but here's the gist, from Sam Henry, Songs of the People, p 286. Have patience; this is a bit long:

"Down in My Sally's Garden

... I have every reason to believe that this is the song o[n] which W. B. Yeats founded his world-famous lyric, 'Down by the Sally Gardens. Alternatively, Yeats may have founded this song on the first verse of another ballad called 'The Rambling Boys of Pleasure':

Down by yon valley gardens
One evening as I chanced to stray,
It's there I saw my darling,
I took her to be the queen of May,
She told me to take love easy,
Just as the leaves grow on the trees,
But I, being young and foolish,
Her then I did not agree.

This song has also been collected in Vermont, U.S.A. ...."

Henry then cites the older "Down in My Sally's Garden." He admits the likeness is distant, and though he doesn't say so, my own theory is that, as poets and songwriters commonly do, Yeats' brain fused the above with the motif of Sally's Garden:

Down in my Sally's garden,
Upon an ivy bush,
At morning and at twilight
There sings a sweet song thrush,

His notes come clearly ringing,
And tidings to me tell,
And oh, I know already
My Sally loves me well.

I kissed her milk-white features
One silv'ry eve of May,
She whispered,
Won't you wander
Until the close of day?'

We wandered in her garden,
The flowers were wet with dew,
I saw the love-light beaming
In her fond eyes of blue.

Down in my Sally's garden,
Where snowy hawthorns blow,
My heart became love-weary
When I at last must go,

The bloom was on the hawthorn
That night I said farewell,
I left my Sally weeping
Down by an ivied dell.

The Wikipedia entry for "Down by the Salley tardens" quotes yeats as writing it was

'"an attempt to reconstruct an old song from three lines imperfectly remembered by an old peasant woman in the village of Ballisodare, Sligo, who often sings them to herself." Yeats's original title, "An Old Song Re-Sung", reflected this; it first appeared under its present title when it was reprinted in Poems in 1895.[2] The verse was subsequently set to music by Herbert Hughes to the air The Maids of the Mourne Shore in 1909. In the 1920s composer Rebecca Clarke (1886-1979) set the text to music.[3] There is also a vocal setting by the poet and composer Ivor Gurney, which was published in 1938; and another by Benjamin Britten published in 1943.

'"Salley" is an anglicisation of the Irish saileach, meaning willow, i.e., a tree of the genus Salix. Willows are known as "salleys", "sallies" or "salley trees" in parts of Ireland.'

Hence the rather thin resemblance is explained by Yeats' small fragment and his vague memory. His poem brought the image to the world. I believe, but am not sure, that there is now a park in Dublin? or elsewhere in Ireland named the Salley Gardens after the poem —please correct me, anyone, if that is inaccurate, it's based on my own vague memory.

Altogether I think this song / poem's relationship to "Rose Connolly" / "Down in the Willow Garden" is nil. But judge for yourselves.

Bob