It means that the event will happen soon; any time from now on - but definitely soon. In fact, "soon" is a concise synonym of "any time now".
"Any time" could be consolidated into one word; however, this is unusual.
If you use the expression, I would recommend "any time now" rather than "anytime now".
As to "now", it does lend meaning to the phrase, by suggesting that though we are not sure of the exact time, we do know that Nana will come soon. Basically, it means any time from now on; probably sooner than later. Without the "now", we would not have any idea of how long we could potentially wait.