After pondering on it don't think it would be as big of an initial changeover for end users as you think.

Theoretically, you could install "Legacy" OS's under a container that works like a virtual server under VMWare/etc. Everything going in and out of that system is converted from/to new standards by the base system, and fed to/from the OS.

For proprietary equipment like a NAS, or something similar where the OS can't just be re-installed, you would need to have a 'conversion server', which essentially looks like a network switch that converts the standard ipv4/ipv6 network traffic to the new standards, and vice versa.

It would take some adjustments to the way networking works on the legacy equipment, but the ways I mentioned could be used to provide that transition from Old Internet to New Internet. Most people/businesses replace their tech within 5-10 years, so as time moves forward, companies phase out the old offerings for the modern tech where the containers and conversion servers are no longer necessary.

This is essentially the way most technology advances have worked over the years. Not sure if you remember PS/2 ports for Keyboard and Mouse plugins? To transition to USB, they included a PS/2 to USB adapter in the box. After several years of that, USB Keyboard/Mouse became the norm. They're now doing that with USB type A to USB type C conversions.