This website has a good explanation of the history of "Come, Come, Ye Saints":
http://fiddle-sticks.com/Liner_Nauvoo.htm#ALL IS WELL - COME, COME YE SAINTS
Its a Harp group that was doing a recording of it in the old "All is Well" style.
The thing to understand is that yes, Clayton was a Mormon, but he was doing the common thing of borrowing a known tune and adapting it when he wrote "Saints."
(Also a good explanation here: http://lds.org/new-era/1975/11/our-heritage-of-hymns?lang=eng)