Concorde,
You said it yourself, "someone from Oxford Training"...
BA are recruiting heavily at the moment but they are taking experienced pilots, not just jet folks, but any reasonable experience. They do not have to set up a cadet scheme (fully sponsored).
The industry is very cyclical... but the over all trend over the last 10 years is conditions have fallen.
If you're waiting for a sponsorship to become available, you might be waiting for a long time and then you might be too old or whatever.