In the US here, but when private and state provided insurance rejects procedures and treatment options one things people can do is get a patient advocate. Apparently this is done to deal with NHS as well.

Have you tried looking at https://www.asthma.org.uk/ to see if you can find some resources to put you on that path? Here people can some times be connected to patient advocacy agencies through some hospitals, too. (The idea is to keep people out of emergency care for treatable conditions)

It took me a moment to recall where I had heard the term brittle asthma before. My grandmother was initially diagnosed with that before getting a better doctor and a COPD diagnosis. Years later she had a nebulizer. I seem to recall that her doctor had to go back and forth with state medical to get it for her.

It's ridiculous that anyone would resist prescribing home nebulizers for individuals that have had to repeatedly use emergency care for nebulizer treatments... If you get an advocate maybe have them include a cost analysis of home nebulizer vs A&E visits in their petition?