1937 with the floorboards definitely has no screed underneath. The floorboards are nailed to battens. These usually rest on a sand bed. Partially, this was/is also ventilated, recognizable by usually small grates covering DIN A6-sized openings on the outside of the exterior wall at floor level. As the owner of an even older house with similar floorboards, I can assure you that you cannot remove the floorboards there without damage.
If the floor is stable, does not wobble, has not sagged or the like, I would leave it as it is. Instead of underfloor heating, you can use fan-assisted convectors in these rooms. (e.g. Daikin Altherma HPW). They operate at 30-35° supply temperature and you can run them without an additional mixer along with underfloor heating in the other rooms via a heating circuit distributor if you want to install a heat pump. Seal the ventilation openings from the outside with mortar, otherwise you'll have cold feet in winter.