Aside from the aforementioned change in management, there's a lot of things that could change.
Primarily, there's a possibility that the Blood War could end in a detente. It's been hinted that it was artificially created. Of course equally compelling is that the Blood War gives Demons and Devils targets other than each other to fight, thus maintaining their respective civilizations (such that they have) in a state other than implosion.
But other targets could serve that purpose just as well. Like the Material plane. Perhaps the war was only to keep the other one from getting to it first, so if either side falls, we're next.
Yugoloths don't count because, logically, if they had the power to stand up to either side they'd do it, and if they had the power to make a play for the material while the other two were distracted by each other, they'd do it.
Or celestia could be a target. Imagine if all the clerics suddenly went through a draught because their gods and demigods are spending all their power fighting off archdevils and demon lords. Infinitely.
In planescape it's been implied the Blood War keeps the demons from invading Sigil. This, for the uninitiated, is a city whose very doors lead to every plane and demiplane that exists. The reason you don't see demons all the time on the material is because they can't just pop over whenever they want. If they had a door from the Abyss to the Material by way of Sigil they'd just make a queue.
Or maybe the other way around. Maybe the celestials would see this as a sign of weakness and attack. Seems a little less likely but not impossible. The question then is 'what would happen if all the demons and devils were killed'?
Can you really kill an outsider? I've heard they just dissolve and form a completely new outsider. Perhaps it would just add Celestia into the Blood War.
Something would change though. I feel like the lords and archs are the ones keeping things as they are. But you have to think of one more thing: Who killed all of them? And what do THEY want? Because if they can pull that off, they are definitely a party to consider.