Welcome to FT, Chris.

UA will give mileage credit only for flights on UA itself, its partner airlines in Star Alliance, and any other airlines with which UA has agreements. How much credit, if any, you receive on partner flights depends on your specific ticket. The flights on the other airlines can be credited to BA or AA, but I doubt that UA would (or could) enter your number. You may be able to enter it when checking in with AA. Otherwise keep all your boarding passes and request credit after your flights are complete.