Well, Q, I think you solved the puzzle. I certainly wouldn't think the lyrics were religious in origin - they seem distinctly secular. The style fits well into that of late-medieval poetry, and the Spanish style of the tune fits well, too. Still, it's not complex Latin. I'm rusty now, but I'm sure I could have written Latin of that quality forty years ago.
Jeremy Taylor is best known for "Ag Pleez Deddy," which I would classify as a mere novelty song, but his lyrics cover a wide spectrum - and many are excellent. I wouldn't doubt that he's capable of writing "O Caritas."

-Joe, son of a Latin teacher-