Try the SCA Minstrel website, and

14th, 15th centuries
R. H. Robbins, 'Secular Lyrics of the 14th and 15th Centuries'
Chambers and Sidgwick, 'Early English Lyrics'
Luria and Hoffman, 'Middle English Lyrics' (with a few tunes)

16th century:
All the broadside ballads that were meant to be sung are listed in the broadside ballad index on my website, as well as the contents of some 16th century ballad and song MSS (Ashmole 48, Rawlinson 85, Cotton Vespasian A 25 [search www.bookfinder.com on 'Peter Seng' for a book on the latter MS. I don't have it, since I have a xerox of the German journal article.]) Some of these ballads are given in Scarce Songs 1, and I have the original of "Fortune my foe" there.
Also in scarce song 1 are songs from BASSUS, 1530, (first published songbook, which also had the music) and Keele's 'Carols', c 1550 (Sole copy now in Huntington Library.

Several other 16th century song manuscripts (e.g. MS Royal Appendix 58, BL MS Addl. 31922 were reprinted in German journals (Flugel in 'Anglia' for last two, Bassus, and a fragment of Keele's Carols) in the late 19th century. I think there's a bit more in the journal 'Shakespeare's Jahrbuch', but I've never seen it.