heckenhocker beat me to it. I understand that you probably feel you were "only five minutes late", but the rules are clear and readily available (and most of the time it's a good thing for the majority of passengers that BA and everyone at LHR is strict about "conformance").

If you'd arrived late because of a breakdown, flat tyre, train strike or blizzard, BA might have put you on a later flight at no charge (especially if you'd managed to check in online before leaving the US). But in your case they can point to the fact that your planned connection was never likely to work, even without your TATL flight not arriving dead on time (which is hardly an unusual occurrence).

Sorry, but I think you'll only waste more of your time if you choose to pursue this.