My guess is that the skynet system has been around for a LONG time. Probably since T2 when they thought they stopped it. They did stop it from being completed, but the programming was probably already written before they nuked the facility. All this time the skynet system from cyberdine has been looking for the links the complete its programming, and my guess is that it just got it from John Henry. Remember when they guy on the phone said he got the scematics from "his brother"? My guess is that it wasn't john henry on the phone, but the skynet system.
I also think John Henry is NOT the part that starts skynet, but the part starts programming for the human resistance. I think he is the father of the "good" terminators.