I think it is inevitable fact, very soon major governments will have their own cryptocurrency simply due to its inherit utilitarian and security purposes.
People can easily exchange it without boarders plus that it cannot be double spent (anti-counterfeits) They can monitor it and freeze if necessary. Forex will exchange government backed cryptos in addition to traditional fiats.
We have to accept it that it is the future of currency and certainly governments will adopt or face being the dinosaur.
What interesting though or rather the question should be is what is gonna happen to bitcoin if that happens? For me I will choose the answer that it can be good for bitcoin, because by then people can easily buy bitcoin using government cryptos (no need for lengthy bank transfer or expensive credit card fees) and by then bitcoin will evolve to full blown digital gold and not as digital currency. Price will stabilize to an ever increasing up trend until it reaches its maximum supply. People will still speculate, manipulate, hedge like what it is now but the general trend is nothing but up because of its finite supply. This is bitcoin advantage over physical gold (gold seems to have an infinite supply).
I think we are going into this direction because government can get both worlds in doing so (taking the advantage of new technology while at the same time maintaining its control over its people).
To predict what the government will do, let us assume that we are the government and lets state what are we going to do. If I am the government I will do this:
1. Ban bitcoin use as form of currency (medium of daily exchange); this is stablishing economic control
2. Introduce government backed crypto currency (centralized); harnessing good benefit of the technology
3. Introduce centralized wallet (aka Banks, ownership is verified); control over people
4. Allow its exchange in forex and crypto-exchanges; freedom to its people
5. Allow people to own bitcoin as form digital gold (not as currency) ; freedom to its people
In this sense I think bitcoin will prevail and here to stay