I can't find it now but the FAQ used to say that NATS were unable to sponsor visas for applicants. You needed the right to live and work in the UK permanently before applying. NATS HR however are the ones to ask about this for a final answer, but I would expect you can't apply until you have the right to work in the UK.
What this does to EU citizen applications at the moment I don't know, before obviously that wasn't an issue but with brexit on the horizon this might change.