I don't know the answer for the case with two horizons (non-extremal case), but when there is only one (that is the extremal case: Q=M in suitable units and common notation for charge and mass), here is how it works.
Let us first note that in the extremal case, the timelike coordinate outside the black hole remains timelike inside it. This is a major difference with the non-extremal case and with the Schwarschild black hole.
Let us look at what happen to probes and observers in this universe.
As you mention, an observer that is at rest in the RN coordinate system and outside the horizon will never see a falling object (say a massive probe, that is a body with a small mass compared to the black hole mass) cross the horizon. On the other hand, it takes a finite amount of proper time for the falling body to cross the horizon. The probe, however, will not meet the (time-like) singularity. It will automatically, at some finite distance of the singularity, "bounce" back and "return" towards the horizon. This is not in contradiction with causality as the radial coordinate is still space-like in this region.
The probe will then again cross the horizon (but this time from inside to outside!), and one can show that all this is achieved in finite proper time.
We thus arrive at the puzzle : where is the probe now ? We know it is outside the horizon. But on the other hand we know it cannot be in the same patch it was before falling into the black hole, as for static observers in this region the probe nerver crossed the horizon.
The solution to this puzzle is simply that the probe has arrived a new space-time patch corresponding to the outside of the horizon.
(This is somewhat similar to the Schwarshild black hole, where you see that there are light rays that seem to come "from" the black hole, but since it is impossible to escape from it, the light rays must come from a new patch that correspond to the inside of the black hole, that is then called a white hole.)
Repeating the above reasoning, we see that consistency of the different viewpoints in the extremal RN universe requires that it contains an infinite number of patches.
I don't know the discussion for the non-extremal case, but I hope that at least some aspects of your question are now clearer.