Welcome to my world! Four years at LHR, BA did the exact same thing to me. They put my luggage tag on the luggage of the guy next to me and vice versa. When we got to Zurich and the problem was discovered, BA knew exactly where my bag was going - to Milan. I was told they would contact Milan and get the bag to me within a day or two. We were staying in Lauterbrunnen for a week which was very small. Fortunately I was able to purchase some clothes and rent hiking boots for a week since my bag never showed up. There was absolutely no one i could talk to regarding this but found using Twitter got some response, like be patient.
BA did reimburse me for things purchase during that week and they were lucky I am very frugal and only spent $500. Upon returning home I filled out paperwork detailing the contents of my lost bag and that is when they took notice.
The biggest surprise of all was that 6 weeks after the bag was sent to the wrong destination, the bag showed up at my front door. There was no phone call, no email, nothing notifying me that bag was returning to Austin. Fortunately everything was inside and nothing lost. The stories it could have told me! Now I am always sure to look at the tag that is attached to my bag to make sure that at least it is going to the same city I am going to.