I've been through LHR several times in the last few months. My anecdotal experience is that the boarding pass still always says Group 1. The two IT systems don't appear to be connected in this respect, which seems to be verified by the fact that every time I am in a situation with BA where I have to prove my status, I have to pull up my status on my phone and show it to the agent.

However, I have asked the agents in the Concorde Room several times and they always says that the gate agents usually will allow you to board with Group 0, but to ask. So, when I go to the gate, I usually go up to the gate agent and sort of play dumb and say, hey, the Concord Room says as a CK I can board with Group Zero, and they said to come up to you at the gate and mention it. Some agents say yes and let me in right then and there, others kind of look confused, but then I push back (still in a self-deprecating kind of way) and I've always been allowed to board with Group 0. One flight they still pushed back somewhat hard that the pass says Group 1, but at the same time let me board with Group 0 anyway. YMMV... Obviously I have to adjust my story if I'm flying a BA flight out of T3 and leave off the Concorde Room part. I'll be flying out of LHR Thursday, so if I remember I'll update it with an additional anecdote if anything changes.