Using ufbx I managed to play the animations as shown here: https://www.keviniglesias.com/videos/GodotAnimationTest.mp4
It seems that Godot is not detecting the full skeleton from the main model (the root bone, B-root, is left out of the skeleton). However, you can work around this by using the skeleton from the animation. Simply drag the mesh to the animation node and assign it the fully detected skeleton from the animation.
EDIT: Uploaded a better tutorial: