Penicillin allergy test; Midlands

[Pottedpalm]

I have been put on the waiting list for a penicillin allergy test. I’m informed that allergy testing for adults is not done st my local (very large) hospital but at Sandwell and that the wait could be long and that there is no contact number for me to enquire about lists ☹️
I’m willing to pay to have it done privately but can’t find anyone to do the test.
does anyone have experience if this, either nhs or private, please?