I doubt if cryptocurrency will replace fiat but these two will instead coexist in the years to come. Most of the fiat used today by every country are their national currency and it is considered one of the important symbol of evey nation. Regulation might only imposed on cryptocurrency exchanges that are centralized and I think it is not possible to regulate cryptocurrency because they are decentralized and can't be controlled.
Cryptocurrency was created to digital transactions, where the way it works is peer to peer so that it does not go through third parties even though it is under government control like the central bank.
This transaction process is very fast, precise and safe. That is on the one hand in theory. But in practice there are many problems, such as bitcoin theft, fraud, even misused for criminal acts.
Therefore, in due course the cryptocurrency will be regulated by each country so that it is not too detrimental if there is a problem. Because something happened on the internet, the government cannot control it.