There's the question of whether or not the rest of the Kingdoms would accept polygamy as a practice anymore from their Royal Family. In the past, I can see them accepting it because the Targaryens had dragons to force the issue but by the present they were gone. If the 7 Kingdoms all more or less decided that they would not recognize Rhaegar's marriage to Lyanna then that's that.
Well, there's no real proof anyway. Everyone who knows is dead. If Jon does become king it will be by right I'm trying to save you motherfuckers from a zombie apocalypse, not right of blood.
There's no guarantee that there's be any Kingdoms levft on Westeros once the White Walkers have made their move, much less a whole seven of them. It's quite possible that the Night's Watch could become the new ruling caste out of necessity rather than any desire for power.