I've run across this blog several times in the past, usually finding it while looking for pig information. But this subject pulled me in enough for me to actually register and post. I live in the States, in southern Illinois with "southern" being the operative word in regards to climate. I know Britain and parts of Europe have recently had a horrendous heat wave, so perhaps readers can relate that our summer temps are like that nearly every day with the added bonus of 40% + humidity. For some time I've been trialing different species in my greenhouse under these conditions. I have eight 5'x10' (12" tall) beds of heavily amended soil in a 48 x 24 double poly greenhouse with roll up sides. With the sides up, vents and doors open, the midday temp will still reach 110+ at chest height, but I can't afford to let prime growing space lay fallow for four or more months.
I have tried many of the plants previously mentioned. Most have not done well, though I've found a variety of cherry tomato (Punta Banda) that has withstood the test for many years, to the point that it has become something of a weed. Okra (all varieties) has done quite well until the aphids find it, then it becomes a matter of determination. Sweet potatoes (any variety, though I tend to go with bush just because of space limitations) dearly love the heat and humidity with the added bonus of being protected from the deer. Cowpeas (bush rather than pole) have done well, but there just really isn't enough room to make it meaningful. Yard long beans (red variety) didn't even slow down when the heat cranked up. Peanuts (several varieties) have also done well, though I put them in there not for their heat tolerance put for the loose soil of the raised beds.