...fairly immediate. The last known dragon dies during the reign of Aegon III (Rhaenyra and Daemon's older son). He's terrified of dragons after watching Sunfyre kill Rhaenyra as a child, so some believe he he had the few remaining dragons poisoned (except for Sheepstealer, since no one seems to know where he went, lol). Well, while Morning's death was never officially recorded IIRC (Rhaena's dragon), there was some talk that the Hightowers poisoned her since they'd lost all their dragons.
It takes about 100+ years, but really, this war begins the death of the Targaryen dynasty, since without the dragons, they're just a bunch of unstable whack jobs. Part of the whole thing is that they truly brought about their own ruin because that's what happens when dragons fight each other. Everybody dies, including the dragons.
I hope the planned show of Aegon's conquest really happens, incidentally. I'd love to see the great Balerion the Black Dread in his full glory.
