The UN publishes a list of recognised member states. ISO maintains a list of codes. IANA uses that list to determine whether a country code TLD is valid. But that's the extent of the UN's involvement in the governance of the root zone.
Aside: Arachnophilia was the first editor I used for web page development in the late 90s as it was installed on every PC in my university's computer rooms. It's amazing to rediscover it now!