I feel like I'm missing something here ... the screen DOES show there's an update available, as a clock icon next to the T in the middle of the top bar on the screen. Touch it and you can start the update any time ...
Now I'm missing something, it says added in Android app October 2015, sure it now notifies you, but I can't see any way to start the update without going out to the car?
@moderator: I don't use android, but according @green1 comment, i'm not sure your can really initiate the update process only with your phone, with is the subject of the feature I suggest... If not, please remove (Implemented) status.