Sounds like the old fasioned Lathe and Plaster method where small strips of wood were used as a baton and the top coat placed on that.
Yes I would be stripping back the wall to find the most solid surface. Just make sure the customer is in agreement and ready to pay for the remedial work.
If no studs exist to take the weight of the rad - you need to create them.
If the house is Grade2 listed the repair is usually with EML (Expanded Metal Lathing) rather than plasterboard. But if its just an old house you can patch with standard plasterboard and flexible filler on the edges. Then skim.
Assume you are looking at something like this?