I suffer from moderate psoriasis but had no problems getting a class 1 (even though I did not go on to a career in flying). The CAA Dr and I had an interesting discussion about it but he did not seem concerned.
I now work in airline operations and have done for many years. Consiquently I have travelled quite a lot. I do find that on long haul trips my skin does become excessively dry, I assume due to the pressurisation.
All the very best !