Skynet wouldn't immediately nuke the world unless it had ready made robots to maintain itself and its weapons. It'd need to keep itself powered, and nuclear winter isn't good for reliable energy generation. It couldn't make new scientific discoveries without the ability to gather evidence and conduct experiments. Computers aren't abstract entities; they rely on our society to function.
It could cause a lot of damage if it was suicidal, but the long game would be necessary if it wanted to outlive us. It might decide to kill us quickly, but it would need to make us totally obsolete before doing it.