This reminds me of a short story I read just a few months ago in Analog. The premise of the story is that all paper on Earth is slowly disintegrating due to the spread of an infectious agent. Researchers are frantically trying to scan and copy all paper books, records, and documents as fast as they can, but don’t realize that they are simultaneously spreading the agent from one library, etc. to another. 
The premise in the OP sounds much worse.
A handful of mathematicians could probably recover essentially all of it. I alone could make a pretty good stab at nearly all of it up to say 1850 and a lot of it up to 1950. Since that time, not that much outside of my specialty and even that fades since about 1975. I have written or cowritten three books and there’s nothing in them I couldn’t reproduce. Not the exact words, of course, but the mathematical content.
I think we could recover a lot of written literature. But other recorded media would be permanently lost. Nobody would ever again be able to listen to the Beatles or watch Casablanca or a Bugs Bunny cartoon.
If compiled code was gone, then the Internet wouldn’t be available. Applications still exist (and work), data files do not. So, modern cars will start up, your phone still works but you’ll have to re-enter all your contacts (good luck getting their phone numbers…).
I’ve written a book where I did basically this to justify shaky knowledge about specific details of the present by people in the future.
All images of things will go, wiping out all art (unless you allow original canvases and film reels and such to survive, which seems unfair if you destroy the books). Semi-accurate images of historical figures and reconstructions of art would be constructed from memory, but the art in particular would suffer. Large swaths of detailed history would be lost, and vagueness would turn up all throughout the historical record - not helped by conservapedia types and other trolls trying to infiltrate and corrupt the new knowledge base at the source. History in general would be reconstructed pretty well by experts and hobbyists stepping forward though.
Math would recover in weeks, pretty much as soon as the experts could get together to discuss it, and science would be reconstructed shortly afterward (perhaps slowed a bit by the problem of shutting out the climate change denier types). If all the computer programs were wiped it would take several months to rewrite everything - adding another year or two if we lost ALL the programs and had to rebuild the operating systems and compilers by bootstrapping up from hand-entered machine code. Most modern tech companies would collapse and new ones would rise in their place.
Losing all the financial information would be a heavy blow and there would probably be massive (and bloody) shifts of power away from people who only had assets on paper. This might interfere with the process of rebuilding The Knowledge somewhat.
If this also fries the computers in cars and such, society probably collapses immediately because our transport system and ability to supply things like food to cities will be shut down. This could, similarly, impede the reconstruction process, possibly fatally, because the people most likely to possess such knowledge are the ones most likely to starve to death when their microwave ovens stop working.
This seems unlikely. Think of all the scientific literature ever created being destroyed in one fell swoop. Sure, we could get back to some first principles pretty quick, but all those studies and experiments that had been performed would be lost. Take, for example, political science and all the polling that’s been done. Sure, we could take new polls, and the basic methodology is sound enough that we could be reasonable accurate going forward, at least once we’ve had another census and recompiled lists of telephone numbers to call, but all the historical data would be gone, and there would be no way to “reconstruct” it. If you wanted to know how Reagan was polling in the middle of September 1984, you’d have a bunch of guesses from people’s memories, but no hard data to refer to. If you wanted to know how the GDP grew in 1945, again, just guesses, no way to confirm this.
Apply this same idea to things like cancer research, or our records of ocean temperatures, etc.
The first body of knowledge to be completely recovered?
All things Python.
*Some people have made the mistake of seeing Shunt’s work as a load of rubbish about railway timetables but clever people like me who talk loudly in restaurants see this as a deliberate ambiguity, a plea for understanding in a mechanized world. The points are frozen, the beast is dead. What is the difference? What indeed is the point? The point is frozen, the beast is late out of Paddington, the point is taken. If La Fountain’s elk would spurn tTom Jones the engine must be our head, the dining car our esophagus, the guard’s van our left lung, the cattle truck our shin, the first class compartment the piece of skin at the nape of the neck, adn a level crossing an electric elk named Simon. The clarity is devistating. But where is the ambiguity? It’s in the box. Shunt is saying the 8:15 from Gillingham when in reality he means the 8:13 from Gillingham. The train is the same only time is altered, ecce homo, ergo elk. La Fontain knew his sister and knew her bloody well, the point is taken, the beset is moulting, the fluff gets up your nose. The illusion is complete. It is reality, the reality is an illusion, and teh ambiguity the only truth. But is the truth, as Hitchcock observes, in the box? No, there isn’t room; the ambiguity has put on weight. The point is taken, the elk is dead, the beast stops at Swindon, Charbrol stops at nothing, I’m having treatment, and La Fontaine can get knotted.*There’s my contribution, anyway. Glad to be of service.
Stranger
There’s science and there’s science. Some science, (like the periodic table and physics equations) will be reconstructed instantly by people who’ve memorized it; physics and nuclear science and optics and the like will have no problems at all, because they don’t really care about the foundational science that led to the current state of things.
Science that depends on historical data to engage in current stuff - well, the science will survive, in that people will remember how to do the science, but the data they’re applying their science to will have been erased, except for generalities (and the odd recalled detail). So science will still resoundingly say that global warming is a real thing, and somebody’ll have at least a vague idea of how popular Reagan was at various points in his presidency, but like with the rest of history, generalities will survive but most specifics will vanish into the mists - and risk being deliberately clouded by people with agendas.
I’m not sure political science is a real science.
I think the real stuff - like biology, physics, chemistry, geology, etc - would be okay. Sure, we’d lose some compilations of data but the scientists working in these fields have already absorbed the key facts that were derived from this data. The same is true about things like engineering and medicine; the people that know these fields could write new textbooks and manuals.
True enough, experts in various fields would not need to re-invent basic principles, but I think that we are seriously underestimating the amount of work that would be required to recreate these various “compilations of data.”
For example, consider something straightforward like the CRC Handbook of Chemistry and Physics. This single volume represents the work of tens of thousands of researchers over the course of the last century or so. It would take quite a while to recreate the data contained in this one book.
There are similar reference books in every conceivable scientific and engineering discipline. Dedicated researchers spent lifetimes collecting this data.
I suspect all of Shakespeare, all of the major holy works, and many of the famous poems–probably even long-form stuff like the Iliad and Paradise Lost. There are just too many people who’ve made a lifetime out of studying much of this.
All of Monty Python, Star Wars, Rocky Horror, etc. of course. But I bet it would contain less modern literature than timeless old stuff, which is a little curious.
The core knowledge would still be there, but all the details would be gone.
Plus with no internet you have no idea what other people know so you can share ideas.
We’d revert to a version of the early 19th century, but we’d advance far more rapidly than they did. We’d probably be more or less back to 1990 levels of technology in 50-75 years I’d guess.
Yeah, there is detailed reference information that could just not be derived from principles; everything from steam and refrigerant tables to mechanical properties of materials; basically, anything that has to be empirically measured rather than just calculated from principles. And some fields like neuroscience or law the knowledge contained in references is so extensive it would be nearly impossible for even a group of experts to reproduce it ex nihilio; I’m sure Kandel, Schwartz, and Jessell have reviewed all 1760 pages of Principles of Neural Science but there is no way they could sit down and write it out from memory, notwithstanding all of the references and diagrams within it.
Stranger
As an aside, did anyone else immediately think of The Knowledge test for UK taxicabs when reading the thread title?
It’s all recoverable, though. Given sufficient time and attention (and there would be lots and lots and lots of scientists itching to give it the necessary attention) every piece of important information in those books could be recreated/rediscovered, restoring the books to a functional equivalent of their pre-wiped states. (Unimportant information would include stuff like “the works cited page”.)
Historical information and creative endeavors, on the other hand, would never reach their pre-wipe states no matter how much time and effort was put into them.
Yeah, but today we have lots of instruments for doing the measurements, accurately and precisely. The biggest problem with any science project is figuring out what you want to learn. Scientists would already know what data they were looking for, and how to find it.
Thousands of college professors would draft their students into “let’s learn by doing” projects.
You aren’t the only one, that’s what I thought at first as well 
The reconstitution of various holy books would be interesting.
Maybe, but I suspect that there would be enough people who who have memorized their scriptures (at least in parts) to rebuild most of them, accurate down to the word, for several different translations as applicable.
You’d lose all the historical annotations, though, and people attempting to argue for improved interpretations based on knowledge of history or older documents would suddenly find themselves adrift without sources. Biblical liberalism would probably gain force just due other approaches being neutered, and the fact that being one of the relatively few documents to be recovered after the purge would give the books the appearance having of divine support to many.