Okay. I put a tarball on Dropbox: https://www.dropbox.com/s/hfditsapv8ngla5/FPS%20tutorial.tar.gz?dl=0
It's basically the FPS tutorial, which I have been using as a base for experimenting.
Scene_Loader.gd is the scene-loading script. Its function load_new_scene_interactively() is called to load a new scene.
resource_queue.gd is the class for loading a scene in a separate thread.
LoadingSprite.tscn is the AnimatedSprite that's displayed during scene loading.
When running the game from the editor on my computer, it only takes a second or two to load a scene, so the issue isn't very noticeable. But when I export the game to my Android phone, it takes five to ten seconds to load a scene, so the stuck animation is more of a problem.