I'm a tad late to responding here but unfortunately this didn't get picked up on my "check how the snap is doing" rounds.
Snaps change the $HOME variable exposed to the apps they run. They're essentially hardcoded to /home/$USER/snap/$SNAP_PACKAGE_NAME/current/ (Where current is a symlink to the internal revision, so 28 in the posts above). There's ongoing work to make this less hardcoded in the future. Regardless, that means that $HOME right now in the snap would actually be $HOME/snap/joplin-james-carroll/current/
Since Joplin writes everything relative to $HOME, the template directory from the perspective of the normal shell would end up being $HOME/snap/joplin-james-carroll/current/.config/joplin-desktop/templates/
Aside from being in the different location, I'd expect it to work in the exact same manner as the official appimage would. The Templates folder at /home/jane/snap/joplin-james-carroll/28/Templates/ is just automatically generated by something else and isn't used at all for the Snap, being symonious with the normal $HOME/templates folder used by various DE's instead.
Tl;Dr $HOME is different from the snaps perspective but everything should work the same relative to it including templates.