If there were a nuclear disaster, it's possible that internet access could be disrupted. However, we're not sure what effect a supervolcano might have on internet connectivity.
Surprisingly enough, the Internet was originally designed to provide highly robust communications even in the event of nuclear attack. I’m not saying that the Internet of today would stand up in the face of such a thing, however.
There are a lot of moving parts that comprise the Internet. While these parts are designed to be resilient there’s a lot of complexity. Design decisions were made decades ago that limit availability and security. I will attempt to provide a high level overview of the moving parts.
There are some things that need to work for the Internet to work well, and those things in broad strokes are: eyeball devices, non-eyeball devices, last-mile connectivity, transit, enabling services, and servers. I won’t cover electricity here except to say that without power generation, the Internet won’t be terribly useful.
In my terms, eyeball devices are the phones, tablets, laptops and other things used to consume Internet content. It’s where people interact with the Internet. Non-eyeball devices is my term for things like IP cameras, “smart” locks, sensors, and so on that use the Internet to provide value. It’s where things interact with the Internet.
Last-mile connectivity is the landline, WISP, cellular or satellite system that you think of as your Internet provider(s).
Transit is not something that most people think about, but these are landline, wireless WAN or satellite systems used by the last-mile providers to talk to other last-mile providers. Landline is the most common by far, wireless is an uncommon use-case, and satellite is unusual for transit (even as it gets more and more popular for last-mile) due to the cost of providing the large amounts of bandwidth required for transit to be worthwhile for most use cases. Think of transit providers as being your Internet provider’s Internet providers.
The enabling services are (in order of importance in my opinion) routing, DNS, NTP, and everything else.
Routing is how networking devices come to a common understanding of the network topology. On my home network, routing is very simple: There is a default route to my Internet router, which in turn has a default route to my last-mile provider. Think of a default route as being directions to “everywhere else.” If there were only one way into and out of your neighborhood, for example, that would be your “default route” to go anywhere beyond the neighborhood.
Routing gets interesting in large private networks, and gets super interesting on the public Internet, especially where last-mile and transit providers work together. The protocol used to route packets on the Internet is designed to be resilient but it isn’t as secure as it could be. Transit providers in particular have had to learn a number of techniques to keep malicious actors from screwing around with routing. Without routing, the Internet doesn’t exist.
DNS (Domain Name System) is almost as critical as routing. It’s function is to translate from words (like equipped.org) that people are able to easily remember, to IP addresses (like 50.57.173.62) that networking devices know how to reach. DNS is distributed and can be made pretty resilient, but has its own security weaknesses.
NTP (Network Time Protocol) underlies a great deal of how routing, DNS, and other things work.
Finally, servers are where the websites and other things that we like to use the Internet for are. Think of “the cloud” and “servers” as being two terms for “other people’s computers that we use on the Internet.”