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Presumably the Poles, already speaking a Slavic language, would have an easier time assimilating the Czechs than the Austrians did
Well, Poles not only spoke a Slavic language, but the one very close to Old Czech to begin with. It's even questionable if they were separate languages back then to begin with. In early XVth century, king Vladislaus II Jogaila (Vladislaus's Jagiellon's paternal grandfather) spoke to Hussite emissaries in Polish and they were able to talk with each other without translator. In late XVIth century (1572), Polish nobility when trying to elect new King, was presented with candidature of Czech magnate, William of Rosenberg (although William himself was rather unwilling to take the crown, being ardent supporter of Habsburg dynasty, trying to sway Poles in favor of electing Ernst Habsburg as king) a lot of them spoke in his favor because he was of "their language" according to them. So I think that if one king ruled Poland and Bohemia, Polish and Czech would stay mutually intelligible for a long time, presumbaly merging in one language.
The big problem is that succession laws are a messy business. It's not always as clear cut as primogeniture, and each crown has both its own laws and power brokers who all want different things. When Casimir IV died, for instance, John Albert succeeded him in Poland, but Alexander was chosen to rule Lithuania. When John Albert died, Alexander got Poland, and after him the succesion in Poland became elective, largely in spite of the will of the Royalty.
Well, in case of Poland, it became semi-elective because of Casimir IV's very ascension. However, the beginnings of election as a way to select new monarch (that's why John Albert was even able to get the throne ahead of Vladislaus) are connected to struggle of his paternal grandfather, king Vladislaus II Jogaila to have his sons accepted as heirs. He struggled to begin with, because mother of Casimir IV and his older brother, late Vladislaus of Varna, was seen as lowborn and was not a Polish princess (Vladislaus II Jogaila got the throne by marriage with St. Hedwig I of Poland, so it was rather important for his sons to have Piast blood in their veins). Still, that struggle was not necessary doomed to fail. He even got Kraków's (Poland's capital) city council to swear a binding oath of this content:
"Nos consules et tota communitas civitatis Cracouiensis recognoscimus publice. Quod nos ad requisicionem serenissimi principis et domini Wladislai Dei gracia regis Polonie et domini nostri graciocissimi, cum consensu et voluntate unanimi reverendissimorum et magnificorum prelatorum, dominorum et baronum eius consiliariorum, fecimus homagium eidem domino nostro illustrissimo et inclito principi Wladislao filio eius et aliis ipsius . Iuravimus eis fidelitatem et subieccionem debitam in hec verba: »Iuramus, quod serenissimo principi et domino Wladislao Dei gracia regi Polonie, domino nostro gratiosissimo, fideles erimus sicut hucusque fuimus, et post ipsum inclito filio eius iuveni Wladislao, habebimusque eundem filium eius pro vero domino et herede, et si, quod Deus avertat, idem filius eius vita excederet, tunc preclare principi domine Heduigi inclite filie eius aut alie proli eius, si quam habuerit, simili modo fideles erimus et habebimus ipsam pro domina et herede. Sic nos Deus adiuvet et hec sancta (crux)«. Et hoc igitur iuramentum et homagium nostrum promittimus nos predicti consules ac tota civitas firmiter observare absque dolo, sicut fideles cives et honesti."
In English, it'd mean roughly:
"Us consuls (city council) and whole community of Cracow (Kraków, contemporary capital of Poland) publicly recognize. That us, at request of most illustrious prince and lord Vladislaus, by the grace of God king of Poland and our most graceful lord, with consent and unanimous free will of most revered and magnificent prelates, lords and baron's, his councillors, we make a homage towards our most illustrious and magnificent prince Vladislaus, his son and every son (of his). We swear than we should swear fealty and obedience in that words: "We swear, that we should stay faithful to our most illustrious prince and lord Vladislaus by the grace of God king of Poland, our most graceful lord, till our deaths and after his death we should stay faithful to his most illustrious son, younger prince Vladislaus, and we accept that son as true lord and heir, and if, God forbid, the king outlived his son, than we accept illustrious daughter of his, lady Hedwig and all her descendants she might have in the same way as true lords and heirs. Let God and His Saint Cross help us. And thus we promise to stay true to this oath, us previously mentioned consuls (city council) and whole community (of that city) with obedience expected from an honest citizen"
That is no small victory. If bishop
https://en.wikipedia.org/wiki/Zbigniew_Oleśnicki_(cardinal) died before the whole affair started, Jogaila would easily obtain what he wanted. Under that law, Vladislaus (king of Bohemia) would automatically succeed his father in Cracow.
Does it have to be Jagiellons? Because the Přemyslids
succeeded, if only briefly.
Swap the fates of Wenceslaus II of Bohemia and Charles I of Hungary, and the rest is coasting downhill.
Well, if earlier POD is acceptable, what about making Pannonian Slavic assimilate Hungarian (a'la Proto-Bulgarian being assimilated by Thracian Slavic, which became modern Bulgarian in turn), as that would make such Slavo-Hungarian mutually intelligible with Polish and Bohemian, which would in turn bolster union's durability.
Granted, there might not even be Jagiellons in such an alt, but there is still an possibility that they will show up (as far as dynastic shenanigans are concerned, they're mostly left to writer free will on this board) and having Polish, Bohemian and Slavo-Hungarian being closely related to each other and mutually intelligible could help in devolping national (not only dynastic conscioussness) - probably named, Wendish, or Vened, or Wenedish (derived from Germanic word "Wend" meaning "Western Slavs). And there is why I think that such Slavo-Hungarian language would be more "Western" than "Southern" Slavic:
Well, to answer that question from purely linguistical point of view (ignoring politics, though they might be influenced by this to some extent + there are usual butterflies on the road).
Pannonian Slavic ( Pannonian dialect of Common Pre-Slavic) most likely (based on work of Jewgienij Helimski, famous Russian Uralic language researcher) some sort of mixed Western-Southern Slavic dialect, which could resemble mixture of OTL Slovak and Slovene - it had features resembling both of them (at least dialect spoken in Blatno principality), but I think assimilation of what would become Slovaks would push that language to a bit more of "northern" side of things. However, there is evidence stating that there was another dialect of Common Pre-Slavic, distinct from mixed Southern-Western Pannonian dialect/language, and that dialect appeared to be straight up identical to what would later become Bulgarian.
The name "Peszt" is an evidence - if the word came from other dialects of Common Pre-Slavic it would be "Pec" and "Peszt" is nearly identical to Bulgarian form of that word.
So, to sum up, how I'd imagine Slavo-Hungarian. First of all, the dialectal divide would be rather prominent in those language. I imagine it'd have two (or three) major dialects - North-Western - the progeny of mixed Southern-Western Pannonian dialect and what would become Slovak IOTL, South-Eastern - progeny of those "Bulgarian" dialect I described before - they won't consider themselves Bulgarians because of receiving Christianity in Catholic Church, which would drive wedge between them an Orthodox Bulgarians to the south of them - in central Europe, national consciousness was often influenced more by religion than language (for example on Polish Eastern Borderlands Ruthenian-speaking Catholics considered themselves Poles, despite not speaking Polish). Croatian might be seen as third dialect of that language or as a still separate language - it won't matter much in the broader scope of things, because of the fact that Croatians wouldn't have any reason to rebel against Slavo-Hungarian state. Croatians in Slavo-Hungary could be compared to Kashubians in modern Poland - there is still an ongoing debacle on whether Kashubian is or is not a separate language, but it virtually doesn't matter because of Kashubians being 100% loyal towards Polish state.
I cannot speak about Czech, but as far as Poland as involved I can tell you something like this. Intially, around time of Ugro-Finnic ATL assimilation, both "Polish" and "Slavo-Hungarian" would be dialects of Common-Pre Slavic, diverging into separate languages. That of course would change over time, but we need to notice several things:
a) Bohemian (very similar to at least "North-Western dialect" from my proposal) still had very high degree of mutual inteligibility with Polish at least up to XVth century - king Jogaila/Władysław II Jagiełło (speaking Polish at the moment) was able to talk to Hussite emissaries (speaking in Czech) without any translator, and I strongly suspect that would be also the case (native speaker of one language being able to speak of native speaker of another language without learning either of them) with north-western dialect of Slavo-Hungarian - who would become main dialect of Slavo-Hungarian and likely core of Slavo-Hungarian literature simply because Pannonia was more populous than sparsely populous and somewhat peripherial Transylvania
b) when the literature and culture of written word in native language arrived in Poland, Polish took a lot of influence from Bohemian/Czech simply because it was written language with longer history of being used for more sophisticated purposes than Polish (though in XVIth century it was still very similar to Bohemian/Czech, when candidature of Bohemian noble William of Rosenberg to Polish throne was proposed, some nobles supported it because according to them, he was of "their language". The case would apply also (to some extent) to Slavo-Hungarian - there are already many hungarisms in Polish, the two among most popular names in Poland - "Maciej" and "Janusz" are corrupted version of Hungarian "Matyas" and "Janos" and imagine how much would they increase if instead of OTL Uralic Hungarian there was Slavo-Hungarian, a lot easier to learn (or in some cases - not even requiring learning) for Pole - so apart from initial similarity of the language, there would be fuckton of loans and loanwords from Slavo-Hungarian in Polish, making those languages more similar than they would be from simple initial similarity.
Both Bohemia and Poland were, at the time, elective monarchies that had had their own parliaments (the Polish Sejm and...I forget what the Czech one was called).
Absolutely false. In Poland, at that time monarch was absolutely free to choose his own heir - Casimir the Great chose Louis of Hungary by that way, although most of nobility didn't like him and didn't wanted him to rule. In Bohemia it was more complicated, but it narrowed down to simple "might makes right". And first Sejm in Poland was convened in (at earliest!) sixth decade of XVth century, it didn't exist in early XIVth century. Bohemian parliament was also called Sejm but I don't know when it arose.