Welcome to the BA forum here on Flyertalk.
The forum's main EC261 thread has a lot of information about CEDR and MCOL, the small claims service. I think this is OK for CEDR, you wil need to highlight the aspects under BA's conrol, which includes ground handling operations at LHR, including the use of remote stands and buses. This won't be a fast process, and one of the things may do is provide a defence that is fairly thorough at first glance, but at second glance may have a lot of irrelevant information. CEDR you can start at 8 weeks, MCOL you can start after you have given BA a reasonable chance to fix the issue (which you have already done).