What does the saying "any time now" really mean?

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.

http://ngrams.googlelabs.com/chart?content=anytime+now%2Cany+time+now&corpus=0&smoothing=3&year_start=1800&year_end=2000

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.

It means "at an unknown time in the very immediate future" - i.e. we don't know exactly when Nana will arrive, but we expect her to arrive very soon, and possibly immediately.

"Any time now" means the event in question is "imminent". That is you know it is about to happen soon, but don't know an exact time.