The oil-wealthy countries aren't really an exception -- even though the countries are wealthy, most of the people in them are poor, because the ruling elite controls the oil wealth and blows it on frippery (e.g. Saudi Arabia) or military adventurism (e.g. Iraq).
When these nations develop a solid middle class (I consider this inevitable, if civilization in general isn't wrecked, though it remains to be seen how long this will take and how much strife there will be along the way), I'm confident that they will follow the usual demographic pattern of lowered birthrates.