I was told they were coke fires by a very old bearded gentleman tending to one when I passed once and asked. As for the logistics of transporting it there, I have no idea.
He was perhaps "having you on"
Coke production is more industrial (and isn't made like that), nor a cottage industry like you saw there.
BTW, have you seen it more than once?, I would suspect it's only a very occasional process?.
Like you say, why bother transporting it there (not exactly an easy place to drive a truck to) when B&Q sell charcoal in bags - mass produced somewhere but I doubt in the Derbyshire Dales.
It has to come from somewhere (although I also doubt it's the Derbyshire Dales) - I would imagine it was being done to use the wood cut down from clearing the paths and grassland, making a saleable product whereas small parts of trees and bushs wouldn't be worth the cost of removal.
Perhaps you weren't aware?, but before the advent of coal in iron making, charcoal was used instead (the invention of using coal instead was made in Ironbridge, Shropshire), and this devastated huge areas of forestland in the UK where charcol was produced, just as you saw in Lathkill Dale.
Maybe there are natural coal seams in Derbyshire. I have no idea of that either. In Lathkill Dale, there are certainly many deep man made tunnels into the rock faces. Just past the crossing in the Dale from the steep road upto Over Haddon.
No coal mines those were lead mines, the area is covered with them - many totally unknown - this is particularly dangerous in many areas of Derbyshire, because many shafts were 'capped' simply by placing planks across them. Over the many decades (centuries?) the soil and grass grows over the top completely hiding them, and the planks rot away - leaving deep shafts covered by nothing but a layer of grass and an inch or two of soil.
You were lucky to grow up in such a nice part on the country. I grew up on the edge of the West Pennine Moors in Lancashire.
Aren't the Moors nice there?, I grew up on the edge of Stanton Moor, at a small village called Birchover.