Oliver
Don't worry too much. The BA medical requirements are stricter than the CAA's for the sole reason that they are sponsoring you. when i joined the cadet scheme, unaided visual requirements were 6/24 in each eye separately (I think these might have changed slightly now). the interesting thing is however, that when i started my course, there were no fewer than 3 people who did not meet the visual criteria set out by BA. The key seems to be that, as was the case for 2 of the 3, that if you already hold a Class 1 medical, and your eyesight has somewhat stabilised (i.e: very unlikely to get anywhere near the +/- 5 D required for a renewal (this rule might have changed as well)).
My recommendation is to contact BA medical services at Gatwick, and see what they say about the visual requirements. If it comes to the worse, don't forget there are eye exercises you can try which really do improve your eyesight.
And don't forget, if you believe you can do it, it will happen. Good luck.