The problem with pagan poetry is that writing reached Western Europe along with Christianity. So, except for some mostly very terse texts written in runes, to the best of my knowledge all literature in Western Europe was written down after Christianity was well established. There may be the odd exception here and there, but certainly not many.

It seems clear that pre-Christian ideas, imagery, etc., survive in many songs, tales, etc., but it's very, very difficult to know what was real and what wasn't. In Iceland, where the conversion to Christianity was not by force, a great deal has survived, but written down later and filtered through a Christian perspective. Nonetheless, there is a good deal of poetry that goes back to pre-Christian times and is accepted by scholars as genuine. Not necessarily singable, though.

There's a kind of gap in between the medieval collections of songs, which outside of Iceland were a matter for the upper classes, and the beginnings of the collection of folksongs and ballads in the Romantic era. There are similarities between the two types of texts but no one really knows what happened in the centuries between. In addition, one should approach the surviving materials and especially the early collections with some skepticism, which has often not been done.

Very, very muddy waters indeed.