If a country switches entirely to IPv6, does this changes anything for the other countries?
(question inspired by this news article)
Definitely. IPv4 has no techniques for talking to IPv6. They are entirely different network stacks, although there are a lot of similarities. IPv6 does have a representation for native IPv4 addresses so they can be routed appropriately. Essentially, the IPv6 country would have limited access from other countries using IPv4. Gateways could enable access to email, web sites, and other services, but this would need to be handled on a roughly case by case basis.
The providers of cable modems and other such equipment are scrambling to be able to provide IPv6 capability. In part this is because they will need to provide dual stacks on hardware with limited memory capacity.
Be prepared to relearn border security, as IPv6 does not use NAT. Different techniques will be needed to secure networks from the external access.
If a country switched completely to IPv6 and still wants to access some old IPv4 datacenters or Networks ressources, it needs the service of NAT64 boxes which can still translate IPv6 to IPv4.
A bit more details about all the transitions to IPv6 tools: http://www.fastlaneus.com/blog/2011/08/19/transition-to-ipv6-at-the-service-providers-nat64-6rd-ds-lite-6pe-6vpe-and-carrier-grade-ipv6-cgv6/
IPv4 and IPv6 are two entirely different protocols, although they do the same thing (actually, IPv6 does a lot more, but that's another story). This means that if someone is using only IPv6, they will not be able to "talk to" someone else whos is using only IPv4 (and vice versa).
In your scenario, this would effectively "cut off" the IPv6-country from the rest of the world and the inhabitants' computers would only be able to communicate inside the country's borders.
In practice, some address translation mechanics would probably be used which could allow at least some traffic to go through. Although it is much more likely they would run both IPv4 and IPv6 for some time to allow the other countries to "catch up" :)