I'm sorry to inform you that most blades were made with almost no heat...
The exceptions are those of the nobles and of the rich. A quite good example are europeans sword"smiths". They could make a razor-sharp, extremely good blade only with a little coal, a hammer, and an anvil, but they mostly made blades from machine-hammered slabs of steel which were later ground by grinders. This way one can make dozens of swords in a day for the same amount of money one'd get for a noble's blade, on which one'd work for weeks. And btw, they had water mills - they used it.
You know, even I could make a good katana blade without modern tools in a month, but instead, I make a dozen messers in a week... for the same money.
This means that a common soldiers blade, everywhere, even in africa, would be made with as little heat as possible, as, you're right, fuel is expensive, while files are not.
And on fancy forge: if you're a noble powerful/rich enough, you can eat ice-cream even in the sahara...