There is one commercial recording, most easily found on LPs and CDs along with spoken-word collections of Tolkien reading his own works. It's by Swann himself at piano and a baritone named William Elvin ("A name of good omen!" said Tolkien), unfortunately Elvin sings in a reserved, fastidious style that doesn't serve the songs well. I've heard a couple of much better live performances over the years, but those were at Mythcons.

I've found a complete performance on YouTube that's closer to the William Elvin style than ideal, and not terrifically performed, but there it is.