I was reading about the life of Jogaila, the first monarch of the union between Poland and Lithuania, and I noticed that the permanence of his dynasty in the Polish throne seems almost a good stroke of luck. From his first wife Jadwiga he had only one daughter who died just a month after her birth. Then his second wife -Anne of Celje - gave him other daughter, who was even betrothed to the future Frederick II of Brandenburg, but died in 1431 before any marriage could be made.
His third wife - Elisabeth of Pilica - died childless. It was only with his fourth wife - Sophia of Halshany - that he got his heirs (Wladislaus III and Casimir IV). And he was already 62 when his fist son from Sophia was born.
So, WI he had been unlucky with his last wife too, and she can't give him surviving heirs? Sophia actually gave birth to other son between Wladislaus and Casimir, but he died after before reaching 10 months, so I don't think it would be too unlikely. What would be the consequences in Poland? Who could claim the throne? And what could happen in Lithuania?
Lithuania and Poland would probably split apart again. Lithuania's royal familly, was as far as I can tell, large.
Poland on the other hand is once more in a bind. Jadwiga was the reason for Jagaillo's claim to the throne, and she was only elected King to screw over Sigismund who had married her sister.
Jadwiga and Mary were both the daughters of Louis the Hungarian (polish name) and Louis the Great (Hungarian name) since he was the King of both countries. Louis himself was the son of Elizabeth of Poland, who was the sister of Casimir the Great, and the daughter of Ladislas the elbow high.
The Angevin claim to Poland (Louiss' house) was based entirely on Elizabeth, but by the time Jadwiga, and even more so Jogailla died, all those descendents were extinct.
Basically, Poland has two options for a new king. They can either go the hereditary route and try and find the Piast prince with the best claim from the time of Ladislas the Elbow high (who died in 1333), and Jogaillo died in 1434, which means they would need to go back a 100 years.
The other option for them, and by far the most likely, is for them to go shop around for a Prince. They would probably pick a neighbouring King, since back then, they actualy liked having their King close at hand (this was the reason they had elected Jadiwaga as King, since they wanted her to be in Poland rather than her father who had been an absentee king).
Now traditionaly, Poland and Hungary were strong allies. They've had a personal union before (under Louis) but they are NOT going to elect the current king of Hungary.
Sigismund, the King of Hungary and Bohemia, was REALLY hated in Poland. They did not like him. Not a bit. But Sigismund is going to kick the bucket soon, and his son in law Albert of Austria only lasts a year since he is killed fighting the Ottomans.
In OTL, after both Sigismund and Albert died, the Jageillons were elected to the thrones of Hungary and Bohemia! But without the Jageillons, Ladislas the Poshumous (of the house of Hapsburg and Albert's son) will likely get the throne a wee bit sooner.
HOWEVER, without the Jageillons in power in Hungary, Janos Hunyadi (father of Mathias Corvinus, considered by some to be Hungary's greatest medieval king) will never come to power!
Thats because Janos Hunyadi fought with the Jageillons to take the throne of Hungary from Ladislas the Posthumous. Janos instead fought for Ladislas the Pole (or Wladislaw III in Poland/I in Hungary). Wladislaw III rewarded Hunyadi greatly for his service, making him a great noble and allowing him to earn a hero's reputation in Wladislaw's wars against the Turks (in which Wladislaw died). Hunyadi's great stance allowed him to become regent for the boy king Ladislas the Poshumous (who having lost the civil war, was made king anyway after Wladislaw died). This later allowed his son to become King of Hungary.
So, by eliminating the Jageillons as a force in central European politics, you create quite the power vacuum in the region.
In short, you could get a number of options. The Poles might re-elect a Lithuanian prince. There were good reasons they chose Jogailla in the first place, and they might do so again. This has a fair chance of happening.
On the other hand, without the Jageillons meddling and rivalry, Hungary might fall into Hungarian hands more permanently sooner, and with it, so might Poland. There was a significant pro-German faction among the burghers of Krakow and other major cities. The Hapsburgs had always had an eye on Poland, and they might really make an attempt for it because of the Hussite problem. Thus you might get Austria, Hungary, and Poland under the same King. Such a situation might also lead to civil war, since there was also a significant anti German faction in Poland who were suspicious of Holy Roman Empire.
Another thing this POD butterflies away is Prussia. When the Order inevitably collapses (which it will by this point), the Poles probably will just absorb it rather let it pass to Brandenburg.