Absolutely impossible to say. Non-State (ie non-NATS) you might be waiting for retirements. Vacancies do occur but they usually want experienced people. I'm not sure how the private schools work but I don't think you would end up with an ATC Licence as you could not do practical work. The NATS scheme isn't exactly sponsored in the usual sense - one joins NATS with a view to a career. They provide training and if one is successful a job is guaranteed. Non-State airfields do sponsor their own staff for training, maybe by NATS or maybe by one of the private establishments.
Download this from thye CAA site: CAP 788: Air Traffic Controller Licensing - A Guide to the Licensing Process. It will provide you with a great deal of information.