I haven't tried this with gmf and LE, but with x-files and DirectX, you can save all the different animation tracks in one model.
So one model would have one track for walking, one track for running, etc.
Then to play these you would call Animate(model, track_walk, ...); and that would play only that animation on repeat.
LE's Animate function has support for different tracks as well if I don't remember incorrectly.
So you'd separate the different tracks in UU3D and save them all into one model. Then you could specify which one of these animation tracks to play in Animate.
*edit*
Just realized I might have misunderstood your post.
You want one file for each animation but without mesh data?
In that case:
I'm pretty sure UU3D can save just animation data. I assume that you'd need a skeleton with the same number of bones and with the same names in all models using that animation.
But I have never experimented with this.
*edit2*
In UU3D you can check or uncheck whatever you want to save (including bones, animations, normals, etc). So to save just animation, uncheck everything else. To remove that information from files just uncheck animation and save whatever else you want to save.