I have no experience at all of those particular clinics - but, if you ring the Anaphylaxis Campaign, the helpline people will direct you to a website which has details of various 'proper' paediatric allergy specialists. I wish I could remember the website address!
We rang the Anaphylaxis Campaign maybe six months ago, about something else entirely (DS seemed to have developed a new allergy to cats, and I wanted to know if we should rehome our cat). We live in a very small provincial city, and the paediatrician we had been seeing was lovely, but was a general practitioner and only did about two allergy clinics per month (presumably with limited funding). I liked her, respected her professional knowledge, and was sad that she couldn't help us more - and I didn't even realise that there was more help available than this. I just thought we were unlucky not to live closer to London or Manchester etc.
After ringing the Anaphylaxis Campaign, and following the advice to be referred (by DS1's GP) to a main allergy clinic which is within travelling distance, we have learned more about DS1's allergies in just half a year than we had learned in the previous five years! He has been tested for all sorts of things, and has passed a peanut challenge.
Travelling some 60 miles to a big city is hard for us, as we are very skint and have had to organise the whole car thing, but it is worth it!!!
I seriously recommend ringing the Anaphylaxis Campaign, whether or not you are a member.