I did that once to a big house built on slab with all heating in the floor. Layed new piping and poured a new 1in, cement floor. I used the type that is self leveling, fireproof and sound proofing . It is used a lot for second flooring to stop noise and ect and works great. I suspect it is not feasible in a small house. Doors are not to much of a problem to raise them.
Also did one where I cut the cement floor and dug down and put new ducking in the floor and new heating system in a closet. Worked great, but again a large house.
I would really look at putting in a new system , above ground. You can usually find a route to put in duct work or perhaps build runs on ceiling or walls that look nice and hide the ducts. Ducts can also be put outside, in attic and ect. You can even put them in the walls in the existing wall system.
Chances are the old pipes in the flooring are copper or steel and the chemicals in the cement use to work on corroding the pipes over time. Most of the old systems put the pipes in direct contact with the cement.
There is also the possibility of installing electric in wall heaters with fans and thermostat. Not the most efficient but with a good insulated house and windows, not to bad.