Floating floor? What will the floor finish be?

Dig out the concrete! Compact new base Add 2 or 300mm EPS, 125mm concrete with grids, battens and your floorboards.

Thanks glad to hear I'm on the right track.

Digging up the concrete and replacing with a limecrete slab and ufh would be the dream but time/money/current heating system don't allow!

Should there be a membrane both below and above the insulation?

Systems like this are available - slabs of insulation to go between the battens, with channels moulded into them to take a thin ufh pipe, and aluminium foil on top to diffuse out the heat.

https://omnie.co.uk/foilboard-batten/
https://www.theunderfloorheatingstore.com/prowarm-foiled-faced-in-joist-batten-system

Not fitted these products myself (yet!) and others are available...

Tapered oak thresholds are available to take up height differences between rooms without tripping over, we had iirc 40mm difference between different parts of our last house, which were taken up by two 20mm tapers at either end of a hallway.

I would scrap the battens and just put down PUR or XPS of suitable compressive strength (both materials are available in 'flooring' grades which are stronger than normal). I think stronger XPS is available than PUR. That avoids both rot and bridging issues.
Membrane on top to prevent air and moisture movement (The concrete is probably just sat on the ground so not all that dry) sealed round the edges: OrconF and/or tape. Then flooring. Either these boards or something which clips together and avoids spreading at the gaps. If you keep the current boards then you may need to cut a 20-25mm batten into the insulation (rout out a space) above the membrane to keep it dry, so you have something to attach the boards to and stop them slopping about.

Tony is right that digging it all out and putting 200-300mm of insulation down there is a much better job (and doing something about the bridge at the wall - external perimeter insulation or cutting in insulating blocks). But that's a lot of work and mess. As you have a convenient batten+board height of 55mm just sticking 40mm of insulation down with 20mm of boards on top (and maybe shaving a few door bottoms to make it fit) would be easy and effective. If you can raise the floor a bit more (would another 20mm really be a problem with e.g. external thresholds?) then that's half as good again (60mm insulation is not bad in a floor - might even be good enough for UFH?).

I wouldn't use PUR because of my idealogical reasons I've explained before. Under floors I'd use EPS rather than XPS because it supposedly can deal with damp better. It's available in a wide range of ratings.

PS Using a jackhammer is good fun if you do decide to dig up the slab, very therapeutic, but as others have said that is more work and mess.

Jackoboard with self levelling will get punctured by table/chair legs, heels etc. I would imagine 12mm cement fibre board supported by insulation and batten would ok, but Id ask Hardiebacker to confirm

I have contacted Hardie to get some thoughts. I assume from what you've said that whilst jacko can be used as a base for tiles it wouldn't be strong enough to support the self leveller by itself?

Thanks. Looks as though tiles may be the way to go instead then. In which case, what are people's thoughts/experience of the jackoboard approach or battens with insulation infill and cement board over?

Hardie have now replied to say that they wouldn't be able to recommend it even with the insulation infill and would need ply/chipboard before laying cement board.

"For this I'd need some kind of self-levelling concrete first."

Not sure that's actually true..

Yes in radon areas, there is a lot of guidance on this.

My view is that radon barriers are not as effective as thought. The radon has escaped through hundreds of meters of rock, often granite. It is a tiny atom that can and does go through polythene! A better plan is to design in a labyrinth with a sump, then use a part time fan to suck or blow the radon out. It is a heavy atom that it tends to sink in air and can be collected and removed safely to outside.

Pro of PIR is better insulation per thickness (approx 1/3rd better than XPS/EPS).
XPS pro is stronger grades, T&G edges to help keep it all flat. EPS is lowest embodied carbon. Not sure if the low-carbon stuff is available in flooring grades. EPS and XPS approx same insulation value - the denser flooring grades tend to be slightly worse insulation.

PUR/PIR is less waterproof than XPS/EPS and may get waterlogged over decades if it's soggy enough down there. If using that you'd want it above the DPC rather than below it, notwithstanding the mentioned 'bucket' risk. That's presumably why kingspan recommend that DPC location.

Floating floor is fine so long as the base is flat enough. If it isn't you will get 'bouncy' spots. Basically add some material to make it flat within 3-5mm before laying DPC and insulation. Self-levelling compound is good. Sand blinding has been suggested too. Presumably also fine so long as the ants never find it.

For the flooring layer the Cellecta gypsum boards seem good screedboard 20 or hideck 18. Immune to water unlike chipboard but no doubt much expense. They also transmit heat very well so good over UFH but you are not proposing that.