If they're on separate bookings, checking-in for both flights will link nothing. In fact, there's probably no way of linking these bookings together at all that will have any effect.

If the GLA-LHR flight on 23 Dec is completely canx and so they don't reach London before 24 Dec and miss the LHR-GCM flight, I would expect that BA will reaccommodate them as soon as possible, subject only to the inevitable mess if there is big disruption all round. It's not because AA has anything to do with it, but because AIUI BA will usually try to do this when it's a missed BA-BA "connection", even on two separate tickets.