Most republics wanted to stay in the union, especially the asian ones. as they benefited from being part of the larger collective. it was Russia itself that clicked the delete button on its empire.
now if they tried to keep the Baltics by force that would be another story.
Even Ukraine, who wanted a lot more self autonomy and control over its own matters ( honestly I am not sure if they would go to war or not, they just might if tanks roll into Kyiv), still was looking to ratify the new union treaty. break ups go bad and the flash point would have been the caucuses and southern Russia.
I don't see the west trying to support anyone but the central government though, too much at stake with nukes and other nasty items laying around for use, it simply would not be in the interest of humanity for things to dissolve into all out multi faction, multi front war. In the end the soviet union was basically confined to the Kremlin.
I think if you want things to go very sour you need to jump back a few years. have the soviets roll the tanks into Poland and eastern Europe to attempt to suppress the independence movements.
this in turn turns bloody, those states inside the soviet union that wanted out rebel as well as they see the central government has replaced Gorbachev and scrapped the liberal policies, also rise up and fight against a crumbling Moscow positon.