Actually, Malcolm, musicologists wouldn't be much use either. What music sounded like and what songs were about are different issues, and musicologists don't care that much about the words! It's medievalists, Latinists and folklorists who have some bead on what narratives were about in this era. Written sources are thin, as several of us have indicated, but songs like The Bitter Withy and The Cherry Tree carol are not such a bad guess after all. Both are derived from tales found in apocryphal gospels, many of which predate King Arthur by centuries. So, although there's no evidence that songs on these themes WERE sung back then, they certainly could have been. The stories were available to people, at any rate. (A good guess from an Arthurian Thieme)!

I would reiterate that one's best sources are the early Irish and Welsh ones. Much scholarship has gone into analysing the links between these early Celtic narratives and Arthurian stories. So, for example, Taliesin is said to be one of the seven men who escapes alive after accompanying Bran the Blessed to Ireland to rescue Branwen; along the way they find and destroy a magical cauldron. This is a Welsh myth, with no Arthurian content. But Taliesin is also attributed authorship of Preiddau Annwfn, the Spoils of Annwn, in which the narrator is one of seven men who escapes alive after accompanying Arthur to a place called Caer Sidi to rescue Gwair; along the way they find and steal a magical cauldron. The narrative similarities are obvious, and both contain the same character of Taliesin. It appears that the Arthurian tale borrowed from the earlier mythological story. (Arthur also frees hostages and raids ireland for a magic cauldron in Culhwch and Olwen, so it was one of the more common stories about Arthur in the eary Welsh tradition. This would be a good guess for a song theme in Arthur's day)

It's quite true that the manuscripts in which these stories survive are much younger than King Arthur, so in that sense it's anachronistic to apply this material to Arthurian times. But most scholars agree that the tales and poems are much older than the manuscripts themselves. On the basis of linguistic data, for example, Preiddau Annwfn is generally thought to be 9th Century, only about 300 yrs post Arthur. That's about as close as anyone can get for sure. And many scholars believe Irish stories like the Tain Bo Cuailgne to be older still--though the manuscripts, once again, are much younger.

One thing I would suggest is to look at the structure of Preiddau Annwfn and other early Welsh poems, particularly those attributed to Myrddin and Taliesin. You'll find that they're not ballad-like at all. Rather, they are mystical and oblique; they make references to stories the audience is expected to know rather than telling stories the audience is expected not to know. It might be interesting to have characters sing songs like that, leading into one of the audience members' recollections of the story being referenced.

So...SOME of us are specialists on SOME of this material, anyway, but as Malcolm says, it's all educated guesswork.