The governments will probably require everyone to register their self-hosted wallets, but they cannot really enforce it, at least not at the moment.
This kind of thing is already well underway. There are protocols such as AOPP to support people KYCing their own wallets. This travesty even gained widespread support from entities which should really know better, like Trezor.
The governments will probably force most centralized web-based wallets and most hardware wallets to perform KYC. But it is difficult to enforce something like that for purely software wallets running on users' computers (i.e. self-hosted wallets). They will probably try to outlaw and/or block the self-hosted wallets, but I don't think they will succeed. At best they can succeed in splitting crypto into a government-controlled, fake crypto and uncontrolled, real crypto. There might be a another problem with the real crypto if its development becomes outlawed, e.g. it may become illegal to develop something like Monero.
So as long as we find a way to convert crypto to/from fiat without going through KYC, then we should be fine.
How do you propose we do this once the government succeeds in getting rid of cash and forcing everyone to move to a CBDC, where they have complete surveillance over every transaction?
For cash it is easy to convert it to/from real crypto, so the question is converting between real crypto and the totalitarian CBDC. Of course I am not suggesting anything illegal, but I can imagine how trade may happen in the future between no too law abiding people.
I can imaging that there is user A who has real crypto and user B who has some CBDC. Then I can imagine that user A creates some random image, that user B likes and buys it from user A, paying user A some CBDC. At the same time some crypto is moved from a crypto wallet owned by A to a crypto wallet owned by B. The random image may contain the actual crypto transaction data but it is not mandatory if both parties trust each other to perform the two transactions (CBDC and crypto) independently.
Obviously this approach has the disadvantage that the government will try to tax the purchase of the random image and charge user A a sales/VAT tax , but there may be other types of transactions that are not taxed by the government, e.g. donations or something else. The tax code varies from country to country.
I can also think of other more complex ways to use real crypto in parallel with CBDC, but the above is probably the simplest to implement.