There are so many thousands of old songs that they can't all be here. This one hasn't been found all that often (though no doubt it was common enough a century ago, and apparently very popular in Devon); the Roud Folk Song Index currently lists two examples only. One of them, noted by Sabine Baring-Gould in Devon and published in Songs of the West (1905), can be seen, with music, at folkinfo.org:

The Cottage Thatched with Straw

Baring-Gould was pretty sure that it was a "published composition", though he didn't manage to track down the original; and no later than the early 19th century.