In case it helps, we often call respiratory consultants chest physicians.
A good way to find a consultant, and most of these tests are common or bssic, is to search for the local NHS hospital's website. They usually list the consultants in each department. Then go to the 'find a consultant' website for any of the big health insurers - BUPA is an example - and enter those names. If they do private practice the telephone number of the private secretary comes up and you can then ring them up and compare prices and availability
I find it hard to believe any private hospital cant do basic pulmonary function tests. However they may not know what bronchial reactivity is.