I'm not sure of BA policies - but I do recall an instance on the DL 82/83 (JFK-NCE-JFK) where my mother held a ticket on the 83 in J, which was canceled for reasons that can really only easily be attributed to light loads (I vaguely remember some mention of there being only 6 pax on the 83, and I can only guess there were not enough on the 82 for them to have bothered operating).

DL booked her into CW on BA via LHR (she was not too keen on this, as she really never books connecting flights unless absolutely impossible), but it was also shortly after the newer (e.g. early 2000s) CW came on the scene, so no doubt the hard product was quite a bit nicer than the average DL product at the time.

In any case, if both outbound and inbound legs are empty enough to make practical a commercial decision of rerouting displaced passengers onto other carriers... I have just about no doubt that most operators would. My only real complaint about the DL handling was that they didn't contact her before coming to the airport. She might very reasonably have elected to fly the next day on DL's nonstop, if she had been consulted before making her way to the airport.