Sorry for dragging up this old thread again, but just wanted to report on my experience with the chat function on the BA website. I had a connecting flight on a BAH cancelled and the re-booked option only left me with a 2.5 hour window to connect from BA Intl to Domestic AA at JFK. While doable, I hate to rush but the website didn't give me any option to make online changes other than accepting the proposed flight. The trip is still a while out but having read about the nightmare of contacting BAH over the phone, I didn't even bother trying.
I used the Chat function which is described in one of the earlier posts and got into a queue straight away. The chatbot asked for booking reference, invoice address and e-mail address and description of the issue. I was 31st in the queue but it went down really quickly and I had a live agent within about 10 minutes., the live agent then only asked for CC details, i.e. CVV/expiry date. I told them which flight I wanted to be booked on and it was all done within about 2 minutes. Really great way of getting things done! For reference, this was at 15.45 UK time on a Wednesday.
Maybe this helps others.