GNOME Music does not index the ~/Music directory directly.
It uses the shared GNOME indexer, which is called
tracker. GNOME lets you configure this in Settings -> Search -> Files. (Select Files and click the cog icon). The dialog shows your Places (xdg dirs like ~/Music), Bookmarks, and Other. You can disable searching in individual Places, enable searching any of your bookmarked folders, and/or manually add folders in the Other section.
This allows you to add an arbitrary set of folders to be indexed for music. Assuming you don't also need it to be a different set of folders than what the file search will index.
tracker status and
tracker info can be used to check the current status of the index.
tracker appears happy to index files outside your home directory, but GNOME Music does not seem to pick them up. That can be defeated by adding symlinks from your home directory.
It looks like album art is cached in some weird fashion. If Music has seen an album before, it may remember the album cover, even if the files you added this time don't include any album art. ("There are only two hard things in Computer Science...")
GNOME Music can also overlook an album in some circumstances, so you may have to remove
~/.local/share/gnome-music to force Music to rescan.
If you have to change permissions on some music files to allow your user to read them,
tracker will not rescan them immediately.
tracker index --file ~/Music does not seem reliable in this situation either, but to trigger a rescan you can just move those files in and out of a temporary directory. Thankfully,
tracker seems able to process files in a reasonable amount of time.