The OP doesn't provide his complete itinerary, so he could be travelling from a third country and only connecting at LHR. He also doesn't say what passport he holds: this is important, because a back-to-back isn't generally considered to be a transfer and BA might deny him boarding at LHR if he is not entitled to enter Bulgaria.
In any case, I'd consider an international back-to-back right now to be foolhardy.