It will become a world currency but whether or not governments will recognize it as such is a different story. Bitcoin at this point is kind of already a world currency, it's known about not just in USA but in 4rd world countries like Argentina, Venuezla, Nigeria (big one) and in parts of Asia that are underdeveloped.
As more and more infrastructure for the population of these countries to GET bitcoin the more widespread it will become. I think it will take a lot of time for bitcoin to finally be recognizes as a currency though, governments don't like to give away their financial stranglehold, they want to be able to control their currency. Once the first big developed country recognizes bitcoin as currency it will be a domino i think.
It's inevitable i think though, see video below from world renowned economists:
https://www.youtube.com/watch?v=6MnQJFEVY7s