ON ILKLA MOOR BAHT HAT was a hymn tune (Cranbrook) used with "Grace 'tis a Charming Sound" or "While Shepherds watched..." before it was a folk song.

Does Danny Boy with "I cannot tell" fit what you're looking for?

As I understand it Vaughan Williams actively sought folk tunes to put with hymns. Many of those hymns in my hymnal give him arranging credit, but clearly say they are folk. The hard part is know how much he rearranged in his arrangements.

There are a lot of American folk hymns that do not necessarily show a link with a non-religious text. Amazing Grace comes to mind. Wondrous Love is another. How Firm a Foundation set to Nettleton.