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the mould will be caused by poor ventilation not just lack of heat from lack of insulation.
It will potentially worsen the problem and not solve it unless you can sort out the ventilation too and think about how to release the condensation build up that will occur if you further seal up the room.
So here is the correct approach:
If moisture from the room can’t escape, it may lead to interstitial condensation (condensation within the structure) or surface mould elsewhere.
Adding insulation boards will slightly reduce the size of the which for a box room may be more noticeable.
Thermal Bridging: If not properly detailed, areas like corners, window reveals, or around sockets they remain cold, leading to localised mould and likely the plasterer is not up to scratch on this when they install it. Also ensure edges, junctions, and around windows are properly sealed and insulated to prevent cold spots i.e proper taping and filling the gaps.
A VCL vapour control layer behind the boards will prevent moisture from getting into the insulation, reducing the risk of interstitial condensation.
You will really need to look at Improved ventilation (e.g., trickle vents in windows, an extractor fan, or a positive input ventilation system) will help control humidity and prevent condensation. Personally with your son’s chest I would look at PIV ventilation without a doubt.
I would also explore more breathable materials like wood fibre or cork boards. These allow the wall to “breathe” and manage moisture better and insulate.
Inevitable mould and condensation like mould can be avoided if the system is really designed and installed properly.
It can so easily be done poorly and you suffer further down the line.
I really do think with your type of home it’s always worth consulting a retrofit professional who can assess the moisture balance in your home and put a whole house plan.
The problem is any measures you make in that room could have a knock on effect elsewhere in the home.
Given your child’s health concerns, improving the room’s warmth and air quality is critical, so whichever route you choose, focus on combining insulation with proper ventilation but the workmanship behind this is really key.
I hope that gives you a solid overview from here on.