Nick Fury certainly would not go to a point in time where World War III was practically about to break out. Leaving aside how difficult it would be to salvage, if he spent too much time mediating a life-or-death duel between two Iron Men, Bruce would probably still cause some trouble for him.
So before leaving, he reset the timeline to when Bruce had just arrived in that universe.
As for Schiller's actions, after learning that the Spider-Man of that universe had vanished, he did not immediately head into the universe to search. Instead, he began reasoning through the matter.
To sort out the sequence of events from the beginning, they had to start with Schiller pulling White Armor over from the future.
At first, since White Armor possessed Tony Stark's intelligence and had no moral bottom line whatsoever, Schiller brought him over to create an engine that could propel the universe forward. Lacking sufficient processing power, White Armor spread a virus to steal computational power from other universes.
Schiller had not placed all his hopes on White Armor either. Or rather, White Armor's universe-propulsion engine plan had been a backup plan. His true target was the Beyonder.
But leaving White Armor there to merely watch over the construction of the engine would have been a waste, so Schiller carried out a series of fishing operations. He got White Armor to target Owen and easily found the Beyonder's location.
There had been an interlude during this period. When White Armor came to Schiller's universe to find him, they had a series of conversations—or rather, mutual probing, Manipulation and Counter-Manipulation.
During that process, Schiller received a message from Amazing Spider-Man through Spider APP. He said that he had killed Captain America.
That "he" had actually referred to Ultimate Iron Man, namely