Well, I don't have the lyrics to the song written out. I looked in OED and saw no appropriate meaning for "cornet" in this sense.

Maybe I am hearing it wrong, and it is "When the corn it was brought home/And placed upon the mow." In those days "corn" meant "grain" or "seed", not the maize that North Americans now call "corn." The song does seem to refer to some some post-harvest festivity though.