This version is the one heard, under "The Siena Waltz" title, on the Alan Kelly Gang/4tet’ Myspace page, and is probably from the pen of Alan Kelly. You can find another version of this under Alan Kelly’s own Myspace page.
I have numbered first bar of each line so as to be more comfortable, as the structure of this tune is a little unusual (48 bars) all the less for me.
I have adopted the key of DMaj according to the first 16 bars, though the tune resolves on E, suggesting Edor. I will correct it if you think I am wrong.
Nice flowing waltz!
I think your first hunch is right, the key is D major.
As a box player I can see no harmony that falls outside of what I would expect from D major - but I am restricted by the chords I have under my left hand; players of cittern, guitar, bouzouki, etc. might have other ideas.
If you were playing the tune through a second time the last bar has a dominant chord (A7), resolving to D major in first bar of the tune.
If the tune ends with the last bar, an open 5th of that E with the A below it allows the tune to resolve in a satisfactory manor.
As an alternative "playing geography", there could be a DC al Fine at the end, with the Fine being at the end of the first or second lines, both of which resolve to D major.
All the best