UK is not a member of Schengen, that's why you need a visa for any Schengen country you visit.
I don't know how an airline would react to that situation, also depends what country you are from. Still a residency is "long" process, first you get 2 years temporary leave to remain, afterwards you get indefinite leave to remain, and after the 3.rd year you can apply for British citizenship, and get a UK passport, so than the problem is solved.
I would assume that there would be some special visa for your situation available, maybe business visa, where you representing a UK company and temporary travelling in and out of Schengen. You should check the UK visa types/regulations.
However I am not sure what requirement the specific airline might have, normally they will have something like the right to work and live within EU, as you might not get a UK base!
But if you going to start doing it, you might as well start with step 1, and take it from there.
I doubt the airlines will want to have to much paperwork to do with this, as there is not exactly a big surplus of jobs available.