Expect lots of IoT or home security devices to malfunction. Or more critically industrial equipment, fire alarm systems, factory automation, etc. A bit over a decade a ago I helped develop an industrial spark extinguishing system. It's still sold and each setup will probably be used for 20-40 years. I know it will suffer from Y2k38, though probably (hopefully) only the event logging. When I raised concerns over this the answer from my boss was "I will be retired by then, don't spend time on that".
But...isn't capitalism supposed to fix this kind of reasoning? Turns out that what you really need is people caring for the right thing to do, and not just for money. Yes, often the right thing to do in a business context will lead to more money. But when time-frames expand and you are not the real owner...
Sure, we say that now, but 584 billion years from now when we use up 64 bits, we’re going to be cursing our ancestors for being so short-sighted.
I imagine a couple of days with the internet misbehaving everywhere is a safe bet.
But my real worry is embedded systems, where very little is 64-bit. Almost every embedded Linux system, IoT crapware, Android anything. Almost none of it will be able to keep time after 2038.
Are other Linux distributions doing the same thing? Sure, and there was a lot of work in the kernel (mostly in the 5.x series) to get it all nailed down. NetBSD and OpenBSD already tackled it.
But "all" can't be guaranteed, because people insist on keeping elevator controllers and industrial processes and anything which hasn't actually had capacitors explode and resistors melt running for an extra decade.
But "almost all" is not "all", and won't be until there. It will certainly be interesting, as stuff will fail in a completely different way from Y2K.