This, above all else, is Jon's priority. And we know from the Season 8, Episode 2 trailer that a good portion of the episode will be preparing for the impending attack.
I predict Jon will put a pin in his identity crisis for now (perhaps he won't even share the news with Dany or anyone else yet?) and that, if any of them survive the Battle of Winterfell, they'll deal with it then.
At the end of the day, I don't think Jon will turn on Dany. Not willingly, anyway. There is the theory that he might need to kill her in order to fulfil the Azor Ahai prophecy and defeat the Night King, though. Which seems more likely than him trying to snatch her crown.