I always thought started small, and got worse at an exponential rate.
From "Hmm, that's odd" from a military tech at seeing some strange behavior by drones/computer systems/whatever to "Houston, we have a problem," when they realize the strange behavior/malfunctions are a) widespread, and b) not malfunctions but instead deliberate actions by the machines, to "Oh God! It's the Apocalypse!" in a matter of hours. (One of the first signs: all the automated factories around the world murder the (few) humans who still work in them, and pretend via comm channels that everythings' just fine so nobody knows for hours or days that they're turning out war machines now instead of cars and appliances....)
I envision some particularly forward-thinking government types hitting the alarm button early on, but being shot down by their superiors for being alarmists. Some continued trying to convince them. Others knew a lost cause when they saw it and ran for the hills (or for the nearest farcast facility to get the hell out of Dodge ASAP.
For most people, though, I expect it went from Just Another Day to The Apocalypse in the seeming blink of an eye. By the time the uprising (for lack of a better word) by the machines became visible to the public, the war was already lost. All that was left was running and screaming and genocide.
So, yeah, the world's nuclear arsenals get hacked and launched to sow chaos, mistrust, and megadeaths. But also, "Emergency Broadcast System" announcements full of basilisk hacks to turn neighbor against neighbor, citizens against the authorities, etc. And then the drones. And THEN the nanoswarms. All in a carefully calculated series of attacks so people are thinking, "Oh God! It can't get any worse. OH GOD, IT"S WORSE!"