I have dust allergies (I used to have LOADS of other allergies, but the good news is lots of us grow out of them! I now (in my 30's) am only stuck with dust). When I was younger, it was bad enough to trigger asthma attacks. Here's what I need to do to manage it:
1) Get an allergy cover for your mattress. It's basically an impermeable barrier that keeps the dust mites stuck inside the mattress. Buy hypoallergenic pillows. Even if the beds aren't really triggering that much, you spend so much time there that this can make a surprising difference.
2) Do you have leather couches or fabric? If they are fabric, and you can, wash the cushions if your kids spend time sleeping on them or lounging on them a lot. I can tell if I've spent too long on our couch, it actually gives me a headache after 90 minutes or so.
3) Any stuffed animals or toys on the beds? Washed or kept somewhere else. (I found that very upsetting as a child, but it really helps!).
4) A HEPA filter can be really useful. Stick it in the kid's room and close the door. Run it as much as you think you need to (we used to run mine all day). You can also buy HEPA quality filters for your vacuum cleaner that might help. Sometimes, even if they don't react to carpet in one place, the carpet in the house might be slightly different, enough to capture the dust mites and keep them there.
My allergies have actually been pretty bad this year (the last month or so). I think it was the super wet year we had, and then all of the sun, suddenly EVERYTHING is either blooming or dropping seeds/pollen everywhere and it is really setting my allergies into overdrive.
It has also taken me about four or five years to get back to my baseline response that I had in the US. Before that, my allergies were markedly worse in the UK, until my system got a better handle on everything. I think I've only had to use my inhaler once in the last year?
I'd keep hassling your GP or check out going private for the diagnosis at least. If you son needs an inhaler twice a day, he probably could use a more managed programme. Good luck!