The damp risk is due to obstructed ventilation of the void under the old floor, and the probable absence of dpm under the later solid floor, and probable leaks from old lead or steel pipes, and broken clay drains and gullies. These are all common, if not ubiquitous, faults in older houses.
Looking on the bright side, the solid floor is probably quite thin concrete, and no match for a modern breaker.
You'd have to dig it out to about a foot below FFL (your intended future finished floor level). A sturdy woman or two can dig it out and barrow it to a skip. Then dig out and renew the old pipes and clay gullies and drain, back filling with crushed stone, bashed down with a vibrating plate. Then sand to blind any sharp fragments, then a plastic DPM, folded up the wall to reach as high as the DPC in the walls like a tray; then rigid foam insulation slabs; then your heating pipes; then your concrete floor (and possibly a smooth screed, or maybe the concrete ground and polished).
This is a good time to lay a new 25mm or 32mm plastic water pipe right out to the stopcock or meter on the supply under the pavement. It will later enable you to get excellent flow for your showers and baths.
The plastic ventilation ducts would probably be embedded in the concrete, depends on the height of the old airbricks and the ground level (which will probably need to be reduced by digging by the sturdy woman).
Fired clay gullies and drains in old houses are always cracked, broken and leaking (I am prepared to believe that there is a old house with perfect gullies somewhere in the country, but I have never seen it, nor met the owner).
Local builders will be familiar with this work, provided they are diligent and experienced. Poor quality work is difficult and expensive to rectify once concrete has been laid on top. In your area the standards required will be copied from templates into Building Regs plans so you can look at them by finding the online plans attached to applications and approvals recorded and searchable on your local council website for extensions. They're a good source of background info .
AFAIK BR's are not required for relaying an existing solid floor but one of the BCOs on here, or your council's enquiries office will know for sure.