It seems at the end of Terminator 3, Skynet relies upon the connected computer network in order to have the intelligence that it has. This made it resistant to any single point of failure to destroy the network.
However, it seems it would make it very vulnerable to a thermo-nuclear war. So far as I know, most server farms (Cyber Centers) are located near major population centers.
It seems highly unlikely that there would be much of Skynet left after a major thermo-nuclear war, because most of the servers would be destroyed, and the links between them would also be destroyed.
Thus, it seems quite unlikely that there could be much left in the way of Skynet. And I'm not even beginning to touch on the subject of EMP, which as the Russians launched their nukes independent of Skynet, would likely have been exploited over the United States and allies, and possibly the world.
How did Skynet then manage to execute Judgment Day, while still retaining enough computers and network to survive?