Yep, for any airside pass, be it an aircraft cleaner, bagagge handler, bird-scarer, cabin-crew or flight-crew - everyone needs to prove what they were doing for the previous 5 years including any gaps between jobs of even a few days.
I am really surprised that you would ever let this concern you when you are supposed to be trying to do your ATPLs. Also, why are you so keen to work for BA and why on earth are you, at this stage worried about if you will get to fly for them? Have you any idea that working for other airliners can be just as fantastic?
Good luck in getting through your studies.