I had a similar situation recently, although it was an AA marketed flight operated by JL (not BA). A frequent poster from this forum helped me author a persuasive email to customer service to get the segments credited. I did have MVPG status at the time.

I think the flight number and fare class are pretty important. In the thread where the OP was denied credit, it was because the ticketed fare class did not map to a fare on the other carrier. Are you able to share the exact fare and flight numbers?