Hard to say, but it's unlikely that fiat will become completely obsolete, but it will definitely become far less important in the future than it is today. It's not possible to completely replace cash in the future. Because there are still some transactions which can only be complied with physical money.
I also think fiat won't become obsolete, but less important. Crypto-Currency will rule the financial system and I don't have doubts about it, the question is if it will be done in a centralized or decentralized way. Crypto-Currency isn't something 100% done and closed, there are many different configurations that can be done with this, resulting in different consequences, but the principle for all the next innovations and technologies on the financial sector are coming from Crypto-Currency.
If fiat crashes, those who are behind this system will find a new way to camouflage fiat system via Crypto-Currency, it will look new, but it will be the same old system...
Becides many people are living in a third world country. A lot of people cannot afford having a computer so paper money cannot be replaced by any digital currency even after how many years.
Even in third world countries many people have already access to internet, in some years everyone will have the chance to access internet, if they wish. I think this is not a limitation. But there will be always some people wanting to deal with physical money, if not for necessity, it can be for whim.
And if the official physical money doesn't exist anymore someday, people will create their own.
In some cities it's a reality, they have their own physical money, useful only on the local stores.