You need an allergy referral from your GP. Isolating the specific allergens will make it easier to work out what may be safe for your dc. I had a few skin prick tests and one big patch test (~120 patches on my back in one go - weird, but not invasive or distressing). Waiting list may be horrendous, so I would also investigate going private if you can.
In the meantime try using fragrance-free products. But your whole household must use them, otherwise your dc will remain constantly exposed to them. IME it took nearly a year to eliminate my known allergens, because everything that hasn’t been used since the previous year was still contaminated, and it can take several washes to get rid of fabric conditioner residue. It also took a few months for my skin to fully heal, and I had to continue using steroid creams until it did.
Products I use:
Personal care:
Simple (but not their hair products, which I find very drying)
Clinique (check - although fragrance free is their thing, they do have some fragranced products).
Faith in Nature and Green People do some very nice ff products online.
Waitrose ditto, often in store.
When my skin is irritated I use a food grade oil on it. My favourite is coconut oil, just the tiniest dab goes a very long way and doesn’t leave my skin shiny. Almond oil and cacao butter are also good, but pure cacao butter is too heavy alone. I melt a small amount and dilute it with almond oil so that it does not set hard again. Always food grade, 100% pure, no additives.
There are ff toothpastes available online. I don’t use them because I need a specialist one. So I have some irritation in my mouth.
Household:
Laundry powder - Surcare (quite harsh, very good for whites and towels), Waitrose own brand (I can’t remember the range, possibly Pure?)
Laundry liquid - Surcare (better for colours IMO)
Liquid soap (for delicates, wool etc) DriPak Liquid Soap (Waitrose or Amazon)
Fabric conditioner -Surcare or Ecover Zero
Bleach - thin, because it’s the only way I can find it unfragranced, but online. Miltons is essentially the same thing, but vastly more expensive. Ecover oxygen bleach granules for laundry.
Washing up liquid - Surcare. Waitrose and Fairy also do ff, but not always available.
Dishwasher - Fairy (online only AFAIK). I’m OK with the regular scent tabs, but not the lemon scent ones if I open the dishwasher while it’s hot and steamy.
Household cleaning - I have gone back in time, got rid of almost everything and use old-fashioned methods. Bleach, bicarb, vinegar, soap, etc. And a handheld steam cleaner.
IMO the best supermarket for fragrance free products is Waitrose.
But it’s just a guess that fragrances are the problem. Quite probable, but unconfirmed. Your dc definitely needs proper assessment and diagnosis.