That is a good question that most of us would have forgotten.
I think you are right in stating that BA wont really want to handle LH as much as LH wont want BA to handle them so I guess it will go to one of the handling agents. Its years since i worked at LHR in fact i think there were no handling agents there at the time. The airlines all handled themselves or were handled by other airlines in which they had some kind of deal.