As a general rule, if your flights were bought on one booking then the more generous allowance will apply. So in this case BA's allowance should apply.

Did you actually buy all flights from BA or a travel agent? In any case as long as the flights are all on one ticket there should be no problems. The other airline will have to honour the higher allowance. Just make sure you have your printed itinerary which will clearly state how many bags you're allowed.