Can I have underfloor heating with solid oak floor?
We have a water based underfloor heating system fitted under wood (old pine I think, not oak) floorboards which works fine.

It will be less efficient than a tile or stone floor as wood has lower thermal conductivity than stone or tile. You would have to fit a more powerful heat source to gain the same temperature gain in your room. It might even be the case that you can't get mats that are powerful enough - the water pipes we have are too hot to touch with bare hands - if you had enough current to produce that amount of heat you might be in danger of melting the wires.