Bush would be obliterated.
It would take him siding with JLU during the fight (if it even lasts this long) or some other superpowered team to even stand the slightest of chances, and this happening is unlikely..
Why am I coming up with a serious sceario for a fight that isn't serious in the slightest?
It all depends on if Bush is allowed to wear a cowboy hat. If Bush gets to wear a cowboy hat, then his powers are limitless. If Bush doesnt get to wear a hat, but keeps the country brainwashed, i still say he wins, because ino superhero is going to kill an american soldier. Bush has alot of powers, i just dont think the hero's can do it. PLus im sure that some of the villains would like to get in arms with bush, and they would help him out too.
I believe it would be one of those rare times when good and evil characters, villains and heroes, unite for the good of the world in order to destroy Bush. Or Doom just goes back in time and makes sure Bush is never born. That or The Flash runs in really fast and kicks him in the groin. Either way I will say woohoo! (Maybe Superman would then chuck him into the sun)
We gotta consider that Post-Crisis Bush still possess the last United Nations Ring, Rubber, Pen and Pencil, also his most loyal herald, Silvio Berlusconi, possess control over power cosmic to a great level, he once took down Bush's evil brother while he was trying to conquer the world.
And don't forget Bush can use his War Nullifier, nuking out all the countries to prevent wars.