With something so major, I think you'd be surprised at how fast things can get done. If we set the world 2-3 years after the Event, it shouldn't be too bad. But I agree, there needs to bea few NPCs n charge of the organization, whether that is through Governmental/UN Liasons, or just having the people who started the organization being Heros, who will lead it and make the assignments to other availible heros, we need to have an easy mechanism for GMs to be able to get the ball rolling with adventures.
Sort of like Matian Manhunter in JLU, he gets information on whats happening, then he assigns teams of heros to go handle it. He might be able to go out and do it himself, but if he did, no other teams would be sent out before he got back, he'd be stretched too thin to do everything, so he sticks to the organizational side of things because thats where he does the most good. Then again we wouldn't need someone with powers to actually do that, just someone that the heros respected and trusted. Possibly a Hero that was injured in a fight, and created the organization to bring heros together to solve problems so they wouldn't end up hurt like he was.