Just to start my answer with a caveat: I'm only familiar with the production of gunpowder in medieval Europe. I can't comment on how the ingredients for gunpowder were extracted in China or anywhere else, nor can I really comment on how production changed after the sixteenth century. With that out of the way, let's jump right in.

As you mention, charcoal is pretty straightforward to make and there was already something of an industry in medieval Europe. Sulfur is pretty straightforward to extract from anywhere you have volcanoes, so in Europe much of it came from either Iceland or Sicily - two islands with quite a lot of volcanic activity to go around. Somewhere like Mount Etna basically just spews out sulfur periodically and you can dig up the dirt that the sulfur has mixed with on the mountain's surface and then refine it down to pure (or pure enough) sulfur.

Saltpeter was the hardest to get. In the fourteenth century, the period when guns first started being used in Europe, you kind of just had to find saltpeter wherever it might turn up. Saltpeter is naturally occurring on decaying matter if certain conditions are met, so it was possible to discover it in the same way that, say, a modern truffle hunter might go searching for black truffles. Gunpowder was very valuable in this century, with gunpowder easily costing more than the guns it was used with, and saltpeter made up the overwhelming share of that high cost.

At the start of the fifteenth century, or plausibly at the end of the fourteenth and we only have references to it from later, the organised production of saltpeter began in parts of Germany. Conrad Kyeser wrote down his recipe for the production of saltpeter in his book Bellifortis in 1405 - and while this is probably one of the most famous medieval manuals for military technology, in the Middle Ages itself it was quite an obscure text. German states very effectively hid the secrets of the production of saltpeter from much of the rest of Europe for at least a century. England only managed to effectively establish its own saltpeter production during the time of the Tudors.

The secret to manufacturing easy and reliable saltpeter was the combination of two ingredients: animal manure and urine. One of the German texts suggests that urine from a 'wine-drunk' man is best, but it's not clear if that is actually the case. There are more factors to consider, the type of earth and the addition of other animal/plant waste can have an effect on the final product. The goal with this process is to create nitrous earth, which could then be turned into saltpeter. I'm not a chemist and chemistry is not my thing, so instead of summarizing how nitrous earth is turned into saltpeter, I'm just going to quote actual chemist and legendary historian Alan Williams and how he describes it:

These authors describe the extraction of salptetre from nitrous earth, using wood-ash (potassium carbonate) to convert the calcium nitrate present into insoluble calcium carbonate; followed by careful crystallization of the filtrate to remove most of the sodium chloride, which remains in the supernatant liquid

The authors Williams is talking about above were English writers in the sixteenth century who knew how to turn nitrous earth into saltpeter, but did not yet know how to make nitrous earth (they knew it involved manure, they hadn't yet learned about the importance of urine to the process).

Over the course of the fifteenth century Germany produced massive amounts of saltpeter, both for use within the Holy Roman Empire and for selling abroad. They didn't have a total monopoly, other parts of the world also knew the secret to making saltpeter - for example there are records of Henry VIII buying saltpeter from Morocco.

The quote is from Alan Williams' article "The Production of Saltpetre in the Middle Ages", Ambix vol. 22, part 2 (1975). pp. 125-33

The classic work on the history of gunpowder is J.R. Partington's A History of Greek Fire and Gunpowder, but it's a little old at this stage and I'd usually recommend people interested in the subject read Bert Hall's Weapons and Warfare in Renaissance Europe: Gunpowder, Technology, and Tactics as an introduction instead.