Yes, we’ve been through the same with our Bedlington. It took years to learn all of the below, so hopefully I’ve saved some leg work!
Firstly, ask your vet to ensure it’s allergies and not a fungal or bacterial infection on the skin. Malaseb is a good shampoo for fungal issues, or the vet can help with bacterial.
I’d also suggest adding Complederm to the dog’s food, as it helps strengthen their skin barrier. It’s a liquid a bit like olive oil and most dogs love it!
Also, don’t bother allergy testing. All dogs are allergic to loads of stuff, so it’s pointless (advice from a consultant veterinary dermatologist). A better way is to try eliminating things one at a time - I’d start with diet (including treats) as the most common allergens are proteins and grains. A lot of dog food says it’s good for allergies but in reality it’s not - I’d suggest Purina HA for six weeks to see if that helps, as it’s truly hypoallergenic.
You could also try changing the washing powder used on their bedding, shampoo, grooming products, etc and see if any of that makes a difference. You can also eliminate dust as an allergen, by using a spray called Indorex on your carpets and soft furnishings - it’s about £13 from Amazon and you do it once every six months.
If none of that helps, I’d suggest you try to get Cytopoint injections for the dog - they block the itch receptors and they’re honestly absolutely miraculous for most dogs.They solved our dog’s itching problems after years of stress, and they work within 24 hours. They do need to be given monthly tho, and it’s around £60-90 per injection, so one for your insurance if you have it!
Your vet will most likely suggest Apoquel, but it’s a horrible steroid which, if used long term, can lead to blindness, deafness, organ failure and all sorts of other nasty things. It also makes the dog drink and pee a lot - sometimes uncontrollably. You can just flat refuse it and ask for Cytopoint instead as it has no known side effects.
Sorry for the essay, but I hope that helps. It can be horrible to watch your dog suffer, but there is hope - ours used to make himself bleed with scratching but now he has no issues at all (largely down to Cytopoint - it really is amazing).
xx
This is a dog that has already suffered a lot and re-homing would only cause her more pain and issues.
I’m glad she’s loved and you aren’t actually considering rehoming.
. He is well socialised and I’ve had him from 8wks and he came from a wonderful breeder, I met his mother, grandmother and great grandmother as the breeder keeps her girls once they’ve finished their breeding years. So I know his history and the line he comes from, all very calm and lovely. As is he apart from this one thing. So I feel your pain and it isn’t just rescue dogs that have this issue. Like all things dog training, it’s baby steps to train them out of it and into the wanted behaviours. Best of luck with your behaviourist appointment 
