JK: Bengel continues rampage, until the Council have enough and put him down. Emperor isn't killed, but that doesn't matter because he'd still be on Voss. Tarnis would probably seal Corsucant away, effectively decapitating the Republic.
SW: Vemrin becomes Baras's apprentice. Since he hasn't got half the potential of the SW, he'd probably die somewhere around Nar Shadaa at best. Baras wouldn't have the clout to beat Vengean, and would be stuck in a sort of limbo, with the Emperor trapped on Voss, but he unable to ascend to the Dark Council and Voice.
Jedi Consular: Rift Alliance never forms. There's no real replacement for the JC, so the Children of the Empire get to rampage for a while before being discovered. Likely by Jaesa, as she wouldn't have been turned to the Dark Side.
SI: Ffon becomes Zash's apprentice, or is eaten by Khem Val. If the former, Zash probably dies before she can get a proper apprentice. If the latter, Ffon either get's killed around Nar Shadaa or get's possessed by Zash.
Smuggler: Pirate fleet never formed, Voidwolf get's to rampage through the Republic.
Agent: Darth Jadus would probably win, effectively taking over the Empire. That said, the Star Cabal would also be a thing, which a nightmare unto itself.
Trooper: The Empire gets to use their bajillion super-weapons they pull out of their ass.
Bounty Hunter: Not an awful lot. I guess Saresh wouldn't become Supreme Chancellor, but they don't really move mountains in the same way.
Overall, the situation would be pretty dire for the Republic. The Empire has far more super-weapons, and the Emperor would be sealed away on Voss. That said, the civil war caused by Jadus might set things on edge.