There are two separate, but parallel events going on at close to the same time, if not simultaneously. The President and the Gallifreyan High Council are causing the events in the End of Time with the Master. In Day of the Doctor, the Doctor contacts the War Council (the military leadership) and effects the events of Day of the Doctor.
There is one line in Day of the Doctor that gives this away. Near the beginning, during the Fall of Aracadia events, Androgar (the aide) mentions that the High Council was in emergency session, that they have "plans of their own". The general replies "To hell with the High Council. Their plans have already failed."
Now it isn't clear whether the events of the End of Time are the "plans of their own" or the plans that "have already failed", but it is clear that they are two separate event lines going on at the same time. I don't see that there is any paradox involved at all.