I don't know anything about Sigismund... and I barely know anything about Matthias... but I did
read a book about the historical Vlad the Impaler at one time so I feel like I'm actually a complete and total expert on the subject matter.
I'm assuming there's no prior diplomatic agreements and if this happened at Robert's Death, then that means the Lannisters will have already been making moves into the Riverlands since it's rather important as a crossroads for everyone and just so easy to invade or was anyways.
Now I have no idea if there's gunpowder in the Riverlands or Westeros... or whatever nor will I care to explore it. I'll just assume those Arquebusiers in the Black Army aren't going to be all that effective after their ammunition runs out because gunpowder could easily be a ggwp type of thing (or maybe not) and I just don't care about that.
This would probably weaken Hungary's rather large River Fleet which has cannons and bombards... but by then they should've smashed any river crossing attempts by the Mountain... and unlike Edmure Tully at Stone Mill (did I mention I've never read the books, only the television show) the Mountain might not escape unscathed looking like a pincushion because those arrows will be partially replaced by cannonballs which AFAIK would be far more devastating to even a big fella like Gregor Clegane.
As a benefit though, the Lannister force under Clegane might be able to flee with less casualties from these opening battles largely because they'll be spending a large amount of time fleeing for their lives and unless the Black Army is on point with the pursuit of their Cavalry heavy forces, shouldn't be completely devastated.
Now the Black Army, as far as the internet tells me, is about 25,000 or more troops which from my Game of Thrones knowledge... is a bit on the smaller side. But the Lannisters had about 30-35,000 troops which isn't a huge numerical advantage. If the Army of the North comes down and Robb Stark and Matthias Corvinus decide to ally up, then they'd actually outnumber the Lannisters. But that isn't taken for granted.
After all there are two other anti-Lannister allies worth considering. The Baratheons... either under Renly or Stannis. And at this juncture only Renly is the one with a sizable army and Stannis is kinda bottled up at the moment. Though being in Renly's camp could be useful immediately in garnering favor with Robb Stark and the Army of the North (or vice versa) while an alliance with Stannis (post-Renly assassination) seems like it wouldn't be too favorable. Stannis would be even more demanding that Matthias bend the knee, and ya know... renounce his Catholic faith sooner or later for the Fire Worshippers... as well as other overt sorcery being practiced... so I feel like a Matthias-Stannis Alliance in such a short term seems rather unlikely.
One thing a Hungarian Riverlands might impact however is how The Reach would react. An Alliance with Renly would deny The Reach forces to the Lannisters obviously, while if Renly is off'd, you'd still have the Hungarian Riverlands with their Riverfleet and powerful cavalry based army just hovering north of the roads leading from The Reach to Kings Landing. This could complicate The Reach sending aid to Kings Landing if they choose to ally with the Lannisters before Stannis attacks Kings Landing.
Tywin Lannister would thus have to deal with the Hungarian Riverlands AND Robb Stark... either independently or allied together and thus... as Robb originally wanted, Kings Landing for all intents and purposes... would be fucked.
There is another option which is Matthias Corvinus could ally with Tywin Lannister but... I feel like he'd prefer working with Robb Stark (who mostly wants independence) more then anyone because he's the easiest to read, the easiest to trust, and the one who'd actually want the least from Hungary at the moment. Plus... the Black Army needs payment and the money is either coming from the Westerlands or the Reach and the former will likely be initiating hostilities right from the get go so... why pursue peace when continuing the War can be far more lucrative?
Ultimately though, you'd have Stannis in Kings Landing and a pretty strong position as well as the Hungarian Riverlands and the North having conflicts in the Westerlands and potentially The Reach. And those butterflies could go anywhere.
