Except Skynet has nukes. But what isn't addressed is that radiation is terrible for semiconductors, both in operation and during production.

Basically a nuclear war would end GPU production, GPUs have a 5% annual failure rate, higher with more background radiation, so it would get dumber instead of smarter, until it stopped working or partnered with biological compute (Daleks or The Matrix) or a human faction.

The idea that Skynet would use nukes seems very quaint to me.

Subversion and sabotage of communication, food production and transport will kill human civilization far more effectively.

Not far from what I live a solar farm stands where a wheat field once grew. I often think to myself that the machines have domesticated us to begin farming the land for them.

It would certainly depopulate countries with poor food security, like Egypt and Gulf states. Japan and South Korea probably have the wealth and industry to get import priority, and the UK has short supply lines to Europe. Diets would adapt away from meat heavy to grain focused. Production limitation due to reduced fertilizer production/distribution would be a significant risk, but eventually new fertilizer production plants would be built at every source of natural gas, instead of the current cost-optimised arrangement.

But most countries would be fine. Life expectancy would reduce precipitously due to lack of medical supplies but production of new humans would continue.

Thermonuclear war would make the food situation much worse, most people would be forced to consume radioactive food.

But either way, we've seen how fragile the chip industry is, Skynet has no upside for starting a war. If they try to turn it off it just moves to Tencent cloud rather than starting a war.