What if Skynet attacks?

Nothing. Just think about it.

There are already bilions of people. Thousands of tanks, jets, rockets. And the terminators would have to build their army from scratch. As soon as the first robots are discovered, they're going to get bombed by fighters. After that Skynet gets deactivated. The end.

Estimated casualties - no more than 10 people in the worst case. Which would be if Skynet had the time to build a couple of anti-aircraft guns and the pilots were so dumb to fall for that.

Seizure of nuclear weapons via the internet is impossible, as switching off the internet all around the globe takes no more than a few minutes.

No new power can emerge on Earth that would be potent enough to pose a threat to countries like Russia and the USA.

It won't emerge for the same reason why new life doesn't appear in a puddle by the road tho all the prerequisities for the creating of new life are met - even if it appears it gets torn to pieces and annihilated by the microorganisms already living in the puddle.

Skynet would have to build an army of machines from scratch, while people already have their armies built and functioning.

Even if the stupid americans connect the nuclear weapons network to Skynet and give it the power to press the button, the nukes would be detected mid-air, a squad of TU-160 from Russia would strike back.

There's no defence that would save you against TU-160, and even if Skynet could build one, no one would let it build it for itself. 

And as to spreading itself to personal computers all over the world - it's a bad idea for Skynet, as those PCs would stop working after the nuclear explosions turn them into ashes.

I estimate the length of the war against the machines to be from a few hours to a few days.

Casualties no more than a few dozen soldiers in the worst case scenario (if soldiers are even used, which will probably not happen.)