itch.io

That should work without having to do anything.

TM (or any other Ren'Py-based game) stores duplicates of savegames in some central system user folder and loads them from there too, so even if you simply extract the new version to a new folder (on Win/Linux/Mac) or update your existing installation on Android, the saves should still be there.

If you're on Android, just make sure to NOT delete the old installation, simply run the new .apk and it should update the existing installation.