The problem with Sumer is icumen in, Caedmon's Hymn, Greensleeves, etc, is that usually we need some indication that a song was available in oral circulation before we call it a folksong. So these songs may be old, but we don't know that they were in oral circulation until modern times.

Looking for evidence of oral transmission sets up some arbitrary limits (how many people have to know it?, etc.) But it's a good rule of thumb for very old texts. Otherwise, you could have an item that exists in one manuscript (like Beowulf) and claim it was a folksong. Looking for such evidence creates a subsequent problem as well: the further back in time you go, the less likely you are to find such evidence, even if the song was in fact in oral circulation. Because not so much was written down, because manuscripts have been lost, etc. So we are left with the older material, saying things like "this might have been a folksong." The degree of certainty you require will heavily affect the answer.

There's another problem with the question, too. Some of the ballads we have in English are clearly what folklorists would call versions of international types. The Scandinavian versions are far older, as the ballad form seems to have existed there first. Thus, does a ballad like "the two sisters" count from its first appearance, or its first appearance in English?

Also, we have some ballads, such as "Bangum and the Boar," ("Wild Hog in the Woods," etc), which are clearly derived ultimately from medieval romances. When it ceased to be a romance (probably read or recited aloud) and turned into a song (sung with a tune) is unknown. Do we count the earliest romance versions? Or (our only other option) take a wild guess?

And John, if you're looking for someone who gives a Rat's Ass, in the folklore world you'd go to what Vladimir Propp would call a "magical donor." Try helping out some old hag on a streetcorner. If you're lucky, she'll turn out to have a magic Rat's Ass she can give to you.