I don't think Spiderman can move in a nanosecond. A nanosecond's warning is a bit short. Time usually "slows down" when the spidersense is warning him of something. Spidersense usually tells spidey what his opponents are going to do before they do it. If he relies on it, he usually stays ahead of them.
Knowing something is going to happen before it happens is the definition of precognition.
I don't think Spidey would have much of a problem dodging Batman's gadgets.