I haven't used a mobile hairdresser since I was about 6 so can't really answer your question directly! However, I have trichotillomania, which basically causes me to pull out my hair, leaving me with bald patches and missing eyebrows and eyelashes. So visits to the hairdresser are terrifying for me, and I didn't go to one for eight years until my sister found me one who knows about trichotillomania.
My advice is to call your local salons and before you book an appointment or give your name explain that you have psoriasis and guage their reaction, so you know how they'll treat you. I'm certain that you'll be assured that it won't be a problem (nonetheless I'd call smaller, privately run salons and avoid salons like Tony & Guys, etc, who seem to employ ... shall I say slightly more shallow people (I'm sure that's a huge stereotyped overstatement, but it's what experience has taught me!)) When I spoke to my hairdresser about the conditions she sees she said none of them were ever a problem (except head lice!) and that people shouldn't be embarrassed. Easy for her to say though, right?! But try it, I was pleasantly surprised.
Mimz