No advantage: if an economist or historian with the better qualities comes along as well, then he will get the sponsorship. Just as much chance with BA as everyone else who has done other aviation oriented activites.
Unless you are absolutely certain that you want a career in aviation, including if you lose your med. then do it. Do it if you enjoy the degree subject, structure and modules, not because you want a job as a pilot. Pursue a degree in an academic subject which interests you and only do this degree if it falls in that category, otherwise you could find yourself making a big mistake.
You'll have as good a chance with BA if you are an outstanding candidate coming from either A-Levels or graduating, and you have the specified qualities. And if you have an ATPL, you cannot apply for sponsorship: you already have what they are sponsoring you to get. I wouldn't narrow your options so much at this point: get a degree in something academic which you have done at school, join the UAS and have a look at the mil. life, and if not for you, apply for a sponsorship at the end.
Good luck