In my experience, only a minority of skin problems with Westie's are related to food allergies. It is more likely to be an inherited disorder similar in nature to asthma and ezcema in human beings.
My reading of the quality literature says there is no cure.
Managing the acute flares is important. Oral cortisone is effective but suitable for short term use only. There is an oral immune suppressant medication available in Australia. It can be effective in at least 50% of dogs suffering Canine Atopic Dermatitis (CAD). You need to discuss whether this is suitable for your Westie with your vet.
I find regular salt water swims at Nudgee beach at low tide also helps and use quality low allergenic shampoos only at bath time.
Tiles rather than carpet may also help.
skin problems