This is not a BA issue, but rather an issue for the operating carrier. Most prohibit "impossible" ticketing or purchasing a ticket without the intent to fly as booked. Needless to say, if you can't fly both tickets, you are violating the contract and subject to cancellation.
Whether the operating carrier cancels one or both or which one if it cancels will depend on the carrier and your (bad) luck.
You have not provided any information as to the route, the carrier or the specifics of the ticket, so it won't be possible to provide any informed information as to whether the specific carrier in question is likely to act. Whether it will has nothing to do with BA or avios.