I have been a welder, machinist, engineer, and someone interested in self-reliance for many years. I read the recent discussion on SurvivalBlog of post-SHTF welding with interest.
I do not disagree that thermite could be made from scrap yard materials (done it, with aluminum filings and black sand (magnetite) from the river), but it would require a custom-made refractory mold for each joint. IMHO two other forms of welding would be much more practical.
Forge welding was the only available process up into the 1800s, and requires only anvil, hammer, fire of coke or charcoal and forced air. Borax or other flux is very helpful on steel, as opposed to wrought iron. Common salt would probably work, too, but avoid the chlorine fumes.
Electric arc welding is infinitely faster and more flexible, and not out of reach post-SHTF.
Engine-driven welding machines are common, and can of course be used as designed as long as their fuels, usually gasoline or Diesel oil, are available.
Gasoline engines can also be run on wood-gas, natural gas, propane, manure gas, with suitable carburetion; Diesels of course on vegetable or waste petroleum oils.
A steam engine burning wood, or a water-wheel, could run a generator welder just as effectively as a modern internal combustion engine, with suitable belting or gearing to provide the right rotational speed.
A medium-size off-grid solar electric power system will also run some small welding machines for limited duty cycles.
Many types of finely compounded welding rods are used today for specific purposes. However, a DC welding machine can be used to weld with coat-hanger or most other types of plain bare steel wire. It is much more difficult to control the arc, and the properties of the weld joint will not be as good, but will be usable for many purposes by a skilled welder. Nor is it out of the realm of possibility to make coated electrodes with better properties, as for instance the coating on one of the most common and most useful modern (E6010) electrodes could be closely approximated with wrapping of newspaper soaked in waterglass, or probably salt or soda mixed with powdered sand.
However I will leave you with a major caveat: If you are not a skilled welder now, then do not expect to do yourself any good by taking up the craft with improvised materials after SHTF.
One of the biggest income-enhancers for the general repair welder like myself, is the guy who buys himself his own welding machine. Fixing it after it was fixed wrong the first time, costs a lot more than doing it right the first time.
As with all skills you may wish you had in an emergency, do not wait for the emergency to acquire them.
Thanks for your service, Mr. Rawles! - Ben F.