As far as I am aware, the last fully sponsored BA cadets were those who passed through Oxford Aviation on a course beginning May 2001 and they graduated around August 2002. There was another BA course behind them, but due to the events of 9/11 their sponsorship was pulled as they were still in the very early stages of the course.
After waiting approximately 6 months or so, all of those cadets who graduated began type-ratings for the 737 and Airbus fleets and are all still flying.
The short-term outlook is not good for any form of sponsorship, but in the medium and long term, as has already been said, anything could happen.