One of the questions I keep coming back to for such a scenario, and still haven't come up with a great answer for, is how does someone living in a world without the ability to manufacture a computer still have computers that work 100+ years after the last one was made? Even manufacturing transistors without modern methods is non-trivial. Will a Z80 last 100+ years? I mean, maybe, if it's kept dry and not exposed to anything corrosive. I've got a Commodore 64 that's ~40 years old and still works...so, 100 years seems reachable, but there have to be extenuating circumstances to get to that "post-tech" world (though I guess in a post apocalyptic world, the value of computers would be seen as minimal for a few years while survival is the only concern, so just forgetting could be enough).
But the book would be more about the consequences of this - do they eventually take the thing apart and jump-start a silicon chip revolution in the 1950s, or (more likely I think) does the government destroy the machine as the UK government did to the Bletchley machines after WWII, and because there's no ground-up computer theory does it set back computing for decades?
So now ask somebody really smart in that technology, like say Jay Forrester who had just finished inventing core memory, to analyze this magic beige plastic box. He could probably recognize that the PCB provided connectivity between parts, but what are the parts, these little flat plastic tiles? I don't think it would be possible to work out from first principles what the functional contents of a DRAM chip is, let alone the CPU. Even if they x-rayed it, supposing they had x-ray tech with enough resolution to resolve a chip, how could they figure out that those little blobs are transistors? Transistors hadn't been invented!
I think they'd have to concede this is "sufficiently advanced" tech, in Arthur Clarke's phrase, to be indistinguishable from magic.
The RAMs and ROMs would be fairly trivial to figure out, as well.
You might not learn the manufacturing process -- that really did take a couple decades of material science and physics advances. But the principles of the machine would be clear. And then you could take the knowledge of that and scale it to the electronics components available in the era. You'd definitely have a head start simply knowing that these things were _possible_, and getting a boost on knowing how a computer could be structured.
Given that knowledge they can try breaking those pieces of plastic apart to see that it's a housing over some sort of ceramic core. Using spectroscopy and chemistry you can figure out what that core is made out of. Now you know what mix of chemicals allows for really high density data storage and computation.
Using x-rays and microscopes they can figure out that the little ceramic die has some sort of structure etched on it. Maybe remove tiny pieces to see what different parts/layers are chemically composed of.
Now they know that there's something interesting about certain elements deposited on top of silicon using some sort of etching approach. Early transistor research was already well along (and had been patented already in the 20s) so it's likely they would have made the connection. Given all that you can start brute forcing industries and ideas around those materials.
You can see both copyright dates, and plenty of other English text. While in 1940 this would have represented incredible futuristic technology, it's pretty obviously made by humans and not a piece of alien magic. It also has components like resistors and capacitors with markings which would have been immediately obvious to 1940s electronics experts.
From Greer's point of view, the factors that make today's hardware brittle are not technical, but economic. Corporations have to make electronics at a profit, and at a price point that is accesible to the average working class citizen. This business model would not be sustainable in the either a fast-collapse or slow-collapse scenario.
Instead, in the novel, governments take over the tech industry sometime in the second half of the 21st century, and treat it as a strategic resource in its struggle to not be left out in the global musical chairs game of climate change + resource depletion. They run it at a loss, and put the best minds they can spare to the task of making a computing infrastructure that is built to last.
By the 25th century, which is the time when the novel's events take place, Humanity has lost the ability to manufacture electronics, but computers built 350 years ago are kept in working order by a cadre of highly trained specialists (most of which have the skills of a geeksquad employee, but still). Common people have maybe heard some wildly innacurate legend about robots or computers. Wealty individuals are probably better informed but still cannot own one of those at any price. They only computers depicted of spoken about are US government property operated at US millitary facilities (or maybe there was one at the Library of Congress, do not really recall, though).
There's one post-collapse hacker in the novel, a secondary character that is part of the protagonist's crew. The author is not an engineering type and dances around the actual skills of this guy, but I'd say he seems able to use a debugger/hex editor and read binaries. His greatest achievement, though, is to fix and set up an ancient printer and recover documents from a disk that was "erased" but not wiped clean.
... you find a controller
... you find their emulation raspberry pi.
... all of a sudden, the world isn't as desolate.
They basically walk you through assembling and programming a full CPU from nothing but NAND-gates in a hardware description language, and in the second part even adding a compiler and higher-level language to the stack.
This, of course, presumes libraries are also mostly gone, since you don't need WikiPedia if you have a library.
The premise isn't that the folks with WikiPedia can rebuild modern society. It's that they literally can't (even if they had better knowledge resources), but would still have a survival advantage from having a little bit of the old knowledge. The fact is that if we lose our modern society, we'll never be able to build it up again. We've dug up all of the easily accessible resources, already. Scavenging from the good old days is the best any post-apocalyptic society can hope for, as bleak as that sounds.
¹ I vaguely recall Wikipedia already provides some way to download all pages in bulk, but I can't seem to find it (if it even exists anymore, or if it ever actually existed instead of me just hallucinating it)
I have libraries and WikiPedia is still pretty useful. Searchability and portability / mobility would be pretty valuable attributes in this type of scenario.
In general, the scenario is, that the whole world is broken down, but full of tech.
So many machines to get back to working. Machines beat muscle on a scale
Most robotics don't work in areas of heavy radiation, because radiation damages electronics.
Even chat is possible for example between two buildings where radio might not penetrate the walls. But sure, at that point if you can lay down cables, then it's simpler to just build a telephone.
There's a nice project page of one here, including an in-depth video about it here. There's a collection of other relay computers here.