Underfloor heating beneath floorboards

As I said, I did not install it myself, but both the timber merchant (sells solid oak floorboards up to 20mm thick) and the installer (did our staircase) see no problem with it. Ok, whether it is ideal is another question. But it is definitely possible!

Precisely glued, classic and "open" planks are placed on a substructure, as the OP writes, and that simply doesn't work... One should also read to the end.

We are also getting room-length Douglas fir solid wood planks with underfloor heating, which are glued directly onto the screed with adhesive.

I would like to have a floor that is as warm and natural as possible, because it is breathable.

Breathable? What is supposed to breathe where?

You install planks on a special substructure, as far as I have seen..

Installation A: screwed onto a substructure or blind floor
Installation B: full-surface bonding with suitable parquet adhesive

You can; you don’t have to. In the US and other countries, variant A is very common.

I would like to have a floor made of wood that is as warm and natural as possible, because it is breathable.

I’m not quite sure what is meant by breathable here. Wood can absorb and release some moisture, but only for buffering; you still have to ventilate yourself. A wooden floor is warm mainly when it is well insulated against the ground. Without proper insulation and sealing, it can be cold and drafty – but in new buildings that is less of a risk.

Wooden floorboards. Maybe even spruce/fir.

Wood is not a good heat conductor. For floorboards, I would make sure not to choose too thick a thickness, rather around 20 mm. Spruce/fir/pine are often thicker because they are less hard and also poorer heat conductors than, for example, oak.
Compared to a floor heating system embedded in screed with tiles on top, the flow temperature probably has to be raised a bit (!), so in combination with a heat pump, floorboards are not optimal, but with a gas heating system, it’s no problem.

But how do you construct something like that in combination with underfloor heating?

Not much different than without. Substructure with insulating mats in the gaps, just not flush with the upper edge, but slightly below. Above that, in the remaining gap, lay the heating pipes not in a coil shape, but in straight runs back and forth (supply and return side by side), and notch out the beams of the substructure at the transition points to the next gap. Use metal plates for better heat distribution. There are ready-made systems for this; sometimes special insulation mats are specified, but usually natural ones based on wood fiber or other natural materials. The search engine is your friend.

That doesn’t work. Period.

If you have absolutely no ... no ... I’ll restrain myself from commenting. Just this much: Yes, it does! Period.

Not much different than without. Substructure with insulation mats in the space in between, just not flush with the top edge, but slightly below. On top, in the remaining space, lay the heating pipes not in a coil shape, but strand by strand back and forth (supply and return next to each other), notch out the beams of the substructure at the transition points to the next space. Use heat distribution plates for better heat distribution.

How funny. At first spontaneous thought I would have also said "doesn't work," but then yesterday during the coffee break I absentmindedly came up with exactly such a construction without ever having seen it. Logical things are always somehow the same.

I was just about to suggest it and lo and behold: it already exists. Well, things exist...

Technical question: Are such floorboards still nailed to the substructure like in the old days? And can't the heat distribution plates simply be one large, thin aluminum plate that also spans over the beams of the substructure? You could just nail or screw it through, right!?

Are such floorboards still nailed to the substructure like in the old days?

Hidden screws in the groove at a 45-degree angle. There are special ones including drilling tips. For example, Assy-Plus floorboard screws.

And for underfloor heating with raised floors, as says, there are ready-made systems. For example, the Thermolutz System Econom-Flex.

There is also a YouTube video including blind screwing...