So a full run down:

Around 5am I called BA (Singapore number) to get this sorted. I was told to ring AA.

I rang AA who tried (apparently spoke to 3 different supervisors). As I said I wanted to get to my destination today there weren't any rebooking options to accomodate that. Apparently it would count as a voluntary change to start at LHR instead of CPH so would be repriced ($8k). I was on the phone for just over an hour as it went back and forth. Maybe AA weren't so worried about the prospect of having to put me in a hotel overnight and pay €600 EU261.

I then called BA again (UK number this number as it was after 6am) who removed the segment from the booking, but couldn't reissue the ticket.

I then called AA who said the ticket is confirmed and the CPH - LHR segment is not there and my first confirmed flight is LHR - BOS.

So long and short of it is I don't know who sorted what and how, but given I'm in a Bath road hotel I'll be heading to Heathrow shortly to make sure that it is all OK. Hopefully it will be, but not 100% convinced.