This game is not distributed by official store (Google or Apple). So all updates for phones must be done manually by downloading app installer. And no there is no update option in game itself. And unfortunately itch.io app is not available on smart phones.
At least for android (and PC) devices progress remains saved and it is continued in new version. If for example you go from 0.5.4.2 to 0.6 than your save should work. 0.2.1 is not compatible with newer version, so if you are able to play this version your progres will not be usable in newer versions.