I've located a library that seems to have the original 1860 sheet music for I'll Twine Mid the Ringlets and I've ordered it. I'll let you know when (and if) I get it.

Meanwhile, you can see a copy of what is supposedly the composer's version of the music online at the University Of Wisconsin Digital Collections, here. It isn't the published sheet music, however, it's a handwritten, undated, unsigned manuscript with the music and lyrics to one verse of "I'll Twine Mid the Ringlets" that was presumably among private papers from the estate of the composer, J.P. Webster. I have no reason to believe it's not authentic, but since it's not the published sheet music there's no way to tell for sure if it's the original music.

Incidentally, as I posted on the other thread, I've done some research and found some additional "Maud Irving" poems identifying their author as J. William Van Namee, using "Maud Irving" as his pseudonym (he was writing for "ladies' magazines" of the time). You can see what I found HERE. As far as I know (please correct me if you know otherwise), this is the first time anyone (since the 19^th Century, anyway) has identified Van Namee as the probable author of the 1860 Twine lyrics.