In cases like this with differing allowances, it depends on the airline that you check your bags in with, at each point along the way. Just because yesterday/last week/whenever you were within the allowance checking in with, for example, AA does not mean you can get the same bags within the allowance checking in with BA.

Note in some places (US springs to mind) where you need to take checked bags through customs at first port of entry, it gets tricky. If bags are redeposited at a central screening/transfer desk then probably okay even if over the free allowance of the next airline. But if bags are redeposited at check in counters then you may find the next airline wants to charge excess baggage (if over their free allowance) despite the bags already being tagged through.