Recycling of anything works along the following lines, and your recycling supply chain needs to work as such.

First thing to understand is that recycling is a specialized form of mining, and, like mining, the economic viability of recycling a given material depends upon:
1) the value of the material being extracted
2) the cost of the mining activity (machinery, manpower, catalytic chemicals, water, fuel, and/or electricity)
3) the pass-on value of the till (that percentage of the raw material that isn't processed)

Step one in any recycling operation is rough separation: In which the pieces of an object to be recycled is disassembled for direct parts recovery. In an automotive scrapyard, this is the part of the process where the car sits at the pick'n'pull station. In third world countries, this is also where plastic is separated from metal, wire from solids from liquids, etc. ALL wire will be held aside as directly re-usable, for fine sorting so that it's easy to find. Wire is hard to make without machinery--it will be as precious as iridium.

Step two is the sort: The disassembled material, now grouped by type, are each sub-sorted. Clean stuff (that can be melted down as-is) from dirty. Copper and cast iron and aluminum from steel. Rubber from plastic.

Step three is secondary disassembly: Circuit boards tested and pulled apart for chips. Dead chips stacked for re-smelting. PCBs stacked for digestion or leech processing.

Step four is cleaning and other prep: Those items which are too dirty to recycle will get cleaned either with solvents or (if metals) by a reduction burn below the point of melting. If it's a metal that will bond with carbon, a secondary process to remove any residual soot may also be necessary, but the first reduction burn should do the trick if it's set up right.

Step five is direct recycling: Those items which can be immediately repurposed as raw material (rather than as jury rigged as replacement parts or as dimensional stock for machining and fabrication; that stuff all came out of the pile in step 1) is sectioned off. Brake drums, for example, are often cast iron, and can go straight to the smelter for re-casting as cookware or for re-allyoing to make tool steel (by the way, your "smelter" can be a guy with a forge or a kiln that's fired by charcoal--this can be done economically at any scale from backyard to industrial with the right setup). Copper , lead, and aluminum will be smelted into ingots.
Please note that most ferrous metals can be directly re-used by a blacksmith, who can re-shape it to suit himself without ever re-smelting. The only reason to re-smelt ferrous metals is for either casting or rejiggering the alloy's chemical properties. Also note that unwelded Wootz damascus can be made in a backyard bloomery (secret was rediscovered a few years ago)--very useful for high-performance blades and other super-hard tools.
Recycling electronics will yield lead, antimony, silver, gold, silica (which in your environment will only be useful to glassblowers and/or to somebody with a micro-forge, and then only sometimes), platinum, aluminum, and copper (all in very small amounts per piece--very high labor cost). Solvents and combustion/distillation can greatly aid recovery in a backyard recycling operation with these things. PCBs themselves are fiberglass resin and epoxy with layers of metal in them, which makes them basically unrecyclable in a traditional sense. These would get thrown in with the plastics.

Step six: Glass. Glass will go to the town ceramicist for making into new glass, fiberglass fibers (if the extrusion tech is available), etc. Un-reusable silica will come here too. Re-usable silica could potentially be made into photovoltaics if you have the mechanical ability to cut the wafers and the chemical ability to do vacuum deposition and whatever else the crystals need (I haven't dug deep into that end of things yet).

Step seven: Plastics. Plastics which can't be re-used are trash. They are essentially un-recyclable, even with today's technology, with a few interesting exceptions:
1) with the right machinery and chemistry you can produce fibers
2) if you successfully sort plastics by type (very difficult, even with current labeling), you can melt it into slabs to use as a material for things like knife-handles, spacers, washers, machining small parts, etc. There will certainly be a demand for some of this kind of thing in your world.
3) Fuel. Pretty much all plastics can be cooked into fuel oil and then into diesel using a basic cracking process called "pyrolysis". This process yields carbon black (which there will be a demand for), fuel oil (important for explosives, as a base for stains and paints, and useful for heating), diesel, and heavy oil slag (which can be used as a lubricant or water proofing agent).
4) Those plastics which cannot be pyrolized and aren't needed for small parts manufacturing can be burned directly to generate heat or electricity (and noxious gasses).

Step eight: Don't forget the value of biotech in your current world situation. Garage biosynthesis means that a sufficiently motivated teenager can engineer algae to do all sorts of useful things that currently require massive industrial processes. If you can figure out a theoretical metabolism path by which feestock A becomes excrement B, you can theoretically engineer a microbe that'll do that job. Currently, synthetic algae exist in the lab that convert CO2, Sunlight, and Water into gasoline, that eat rust and excrete iron and O2, that digest plastics into basic monomer proteins that don't poison the food chain, etc. Naturally this also means that some of your bad guys might get the idea to make bioengineered weapons that target materiel instead of other life forms, which could get interesting.

Remember that each step of the recycling process will have till, and where there's a pile of relatively uniform raw material someone somewhere is going to figure out how to take that trash and turn it into treasure, limited only by the available tech, the constraints of economics, and the laws of physics.