The problem with the Middle Ages is that they didn't speak English - at least, not English as we know it. Chaucer's Canterbury Tales, written in a fairly late form of Middle English, begin:

Whan that Aprille with his shoures soote        
The droghte of Marche hath perced to the roote,        
And bathed every veyne in swich licour,        
Of which vertu engendred is the flour;

('soote' = sweet; 'swich' = such; you can probably get the rest)

Orally, from what we can tell, the language was even more alien to a modern ear: that first line would be pronounced something like

Hwan THat ap-REEL-eh wiTH hiss show-ress SOH-teh

(the capitalised 'TH' stands for the 'th' sound in 'thing', as distinct from the soft 'dh' sound we use now)

You get the picture: Middle English sounds strange. To prove that a song was a mediaeval survival you'd have to have a Middle English version alongside a version in sixteenth-century English (which looks pretty odd, but nowhere near as odd as ME); I can't imagine a song that would qualify. The best example I know of a song from the Middle Ages is "A maiden that is makeles" - and that doesn't qualify for your purposes, because the poem survived in written form (in Middle English) and was set to music much later. I wrote a bit about the difficulty of rendering this song faithfully here.