The Deluge in Poland and Lithuania, which was a crisis period extending from the early 1650's to the early 1660's, saw the Polish-Lithuanian Commonwealth be invaded on all sides by Sweden, Russia, the Cossacks, Brandenburg (to a small extent), and the Duke of Transylvania. There were also some Lithuanian separatists. Some well-timed tactical victories by Poland and distractions by the occupying powers (such as Sweden getting embroiled in a war with Denmark in the middle of it) saved the Commonwealth from utter collapse.
But what if the Commonwealth was destroyed during this time period, and partitioned by these invading powers? How likely was this?
The Treaty of Radnot (1656), for reference, allocated:
-Ducal Prussia, Masovia, Samogitia, and Inflanty (Livonia) to Sweden
-Greater Poland to Brandenburg
-Lithuania as a Swedish vassal (which could foment future conflicts with Russia)
-Podolia and Volhynia to the Cossacks
-Lesser Poland to the Transylvanian Duke, George II Rakoczi
What would have been the effects of an earlier "partition of the white eagle"?
Ducal Prussia would become a Swedish Vassal but still under the rule of the Hohenzollern, Brandenburg would also receive Ermland (which would be integrated into Ducal Prussia).
I suspect that the Swedes would annex Royal Prussia (the Polish Corridor).
So here's my thought Sweden would start a force conversion of their territories to Lutheranism, through they would likely extend tolerance to the Reformed/Calvinists. I suspect it would be pretty easy in the Roman Catholic areas, but it would likely be far harder in the Greek Catholic and Orthodox areas. I suspect Royal Prussia will become majority German speaking. Lithuania as it become converted will likely end up funding the Swedish state.
The Elector of Brandenburg would be a a vassal of the German Emperor in his HRE territories, a Swedish vassal in Ducal Prussia and a sovereign Prince in Greater Poland (where his precise title would be unclear, I'm not sure what the title High Duke, which the dukes of Greater Poland has used was, maybe it would simply be translated as Grand Duke). He would begin to convert the people of Greater Poland and Ermland to Calvinism and Lutheranism and settle Protestants in his new territories. The result could either be that the Poles of Greater Poland became Protestants or that Germans would end up the majority. A interesting fact would be that instead of Prussia, the Electors of Brandenburg would likely end up being "König in Grosspolen" instead of being "König in Preussen". A major goal for the Hohenzollern would likely be to end their vassalage to Sweden in Ducal Prussia and gaining Royal Prussia (maybe in alliance with the Transylavian which could go after Masovia).
No idea how a cossack state would end up.
The Duke of Transylvania would likely be crowned Polish king, he would favour Calvinism, but likely set up a large degree of tolerance as already existed in Transylvania for all Christians. In the longer term he and his successors would likely turn against their Ottoman overlord, and I could see them replacing Poland at Vienna, they could use the following war to expand and gain control over the Romanian Principalities and the trans-Danubian Ottoman territories. The results would be a new Poland which was pretty much made up of Austrian 3rd Partition Polish territories and Interbellum Romania, of course such a kingdom would likely expand south as the Ottomans weakens, it would be ironic to have a Poland without Warsaw but with Constantinoble.
As a POD for why Denmark doesn't enter the war, I would say Frederick III of Denmark falling down a bunch of stairs in 1656. Thsi would leave Denmark under a regency council unwilling to start a war.