I've recently been reading a fun but not first-tier mystery novel in which the English folk tune "Bushes and Briars" features. I'd never heard it before, so I looked it up. Apparently it's very famous. I think this simple rendering by Julie Christie in the movie Far From the Madding Crowd makes it easy to pick up the tune.
Now listen to this Advent carol.
Similar, no? (HT to Eldest Daughter for noting the similarity.) In fact, strikingly so. The tune for "The King Shall Come" has various names (this site seems to be calling it "Consolation") and is an American folk tune. It appeared in the Kentucky Harmony published in 1816.
When it comes to simple modal tunes, it's very hard to be sure when there is a causal line and when one simply has a case of independent discovery. But I wouldn't be all that surprised if it turned out that someone (or several someones) carried the tune "Bushes and Briars" to America where it morphed into the tune later picked up and used with the words "The King Shall Come."
Both very lovely.