I've been retired for about 5 years now but I can remember, some time before that, BA issuing an edict "no through check on ANY mutiple tickets". Period. The little baggage paragraph only opens a small and specific chnink in that statement.
I still don't know what would happen if an inbound delay caused a misconnect. We were nearly an hour late out of T5 but had a fast crossing arriving nearly on time. That and a retime of the LAN flight to about 40 minutes later meant we were OK but I did wonder.