I was on the Madrid Passenger's end of this kind of situation a couple years back.

I flew BA into LHR and by time I processed through immigration and got to the baggage carousel, the bags from our flight were circling. Mine was not there; but there was one almost identical to mine traveling round and round.

I waited until all bags seemed to have been claimed and that one similar to mine was still there. So I took it to the baggage office and explained to the agent that I thought there was a good chance that the owner of that bag mistakenly picked up mine.

The agent confirmed that mine should have been there, looked into the reservation of the straggling bag's owner and called her mobile number.

The perpetrator was in a cab well on her way home somewhere in Central London. She realized her mistake, apologized profusely, and offered to have the cabbie take it directly to me at my hotel at her expense as soon as he dropped her off.

She did that. My bag was waiting for me at the hotel when I arrived.

I recognized that we all make mistakes and this lady owned hers and did the best she could to make it right. I was fine with it. (Of course, it didn't hurt that all this meant that I didn't have to schlep my large bag onto the Heathrow Express and then into a cab to my hotel. )

That experience seems to me a model for how to someone making a mistake like this might handle it. If I were in the OP's situation, and I couldn't deduce the owner's phone number from a luggage tag, I'd ask BA to call the owner and explain that I would be happy to courier their bag to them immediately if they would be so kind as to call me with their delivery details. I'd send it out straight-away with a little something in the side pocket as a token of my apology.