The way I view it, Bitcoin is already a world currency. I do know how imagine it to be, but it won’t get to that extent. For now, since you and everyone around the world is able to make use of it, and you can send money and also receive from anyone, then it’s already a world currency.
You make use of Bitcoin and the government does not charge you any tax (except if you use a regulated exchange), and your transaction is always peer-to-peer, then what else are you expecting? If you mean whether the government will accept it, no they won’t accept it, rather they would create their own cryptocurrency.
technological development without the support of the government I think will hamper its growth. although bitcoin is a global currency, and we are free to transact with people anywhere, but in reality to meet daily needs, we have to exchange it at local exchange. and that is subject to tax