When two or more Doctors cross the Doctor's personal timestream only the most recent one will remember the events that took place. The War Doctor remembers up to the point where he made the conscious decision to use the Moment.
However, when the Bad Wolf Rose appears, the Doctors' timestreams have already started to cross each other. And in fact the remainder of the War Doctor's experiences in "The Day of the Doctor" are related to the Eleventh Doctor (and the Tenth Doctor). So, all he will remember is making the decision to use the Moment, and then the next thing that he will remember is regenerating in his TARDIS, after Gallifrey and the Time Lords are no longer part of the Universe. This is more or less explained by the Eleventh Doctor just before War departs.
This explanation from "The Day of the Doctor" however contradicts earlier instances of similar events. In "The Five Doctors" both the First and Second Doctors remember the events from their earlier multi-Doctor story "The Three Doctors". It is only when Rassilon wipes their memories of both events that they cease to remember them. ANd the Tenth Doctor clearly remembers the events of "Time Crash", when he met the Fifth Doctor, from the Fifth Doctor's perspective.