So this is a political problem -- here's my prediction of what will happen (given that we live in the real, political, world).

A political entity -- not necessary a sovereign government, but perhaps a bank or financial institution -- will offer a currency swap to existing blockchain holders to adopt their crypto currency. The inducement will be a limited time window to put in your claim, with all unclaimed but mined numbers going to the financial entity to reward their followers or stakeholders.

In the real world, this is called escheat and it is a power of the crown. Bitcoin is essentially a system for recording deeds to digital land. They aren't making more numbers, so the problem is the political resolution of competing claims to the same resource. This sort of claim comes, in the end, to a network consensus of who is the sovereign.