A follow-up to my own post: The Richard Hill manuscript is quite a source of songs. I ended up including entries for 32 of its poems in the Ballad Index. A lot of these are things that I believe were traditional then but do not appear to have survived in tradition until modern times, but there are several that either did survive or that have been taken up by modern composers. Among these are

A Babe Is Born All of a May
The Boar's Head Carol
The Corpus Christi Carol
Gabriel of High Degree
The Holly and the Ivy (or, at least, one of the many holly-and-ivy songs of the period)
Joly Wat
The Nut-Brown Maid
O Mary Mother
Of a Rose, a Lovely Rose
This Endris Night