The question of whether someone is on strike or not is completely different from the question of whether someone is sick or well. You can be on strike and sick or on strike and well. In other words, whether you are sick or well is not relevant to the question of whether you are on strike. I understand therefore that BA is legally entitled to assume that anyone who doesn't show up for work is on strike, since whether one is sick or well is not relevant.
I'm not saying BA won't make retrospective exceptions for people with a doctors note. But I think this might only be relevant if your period of sickness started before the strike days. I'm fairly sure if you called in sick - say - on Sunday for the first time, then it will be reasonable to assume that you are on strike.
In any case, if you're really not on strike I'm sure you can get into work and it will then be obvious you are unfit to fly. If you really are too ill to drag yourself into work then I guess you'd be in hospital anyway - in which case I'm guessing BA will let you off!