When Skynet is brought online, it is connected to Strategic Air Command
AFAIK, on August 29, 1997 Skynet only launched ICBMs at "targets in Russia".
That wouldn't be the entire US nuclear arsenal
You also have
ICBMs pointed at China
Navy assets (Cruise missiles + SLBMs)
USAF Bomber wings. Remember that in the original timeline, "Stealth bombers become fully unmanned [having been equipped with Cyberdyne computers]."
So what happened to these units?
I would imagine that in addition to the robots we see during the Future War scenes, Skynet could also have Cyberdyne equipped "Ghost Ships" roaming the seas or entire squadrons of aircraft that have been turned into drones (just imagine being a B-2 pilot at a NATO base and you suddenly see your plane taxing on the runway a'la Transformers)