I can only confirm what’s been said: BA’s policy in such circumstances is to not through check luggage even on consécutive but separately booked BA itineraries.

”Linking” itineraries makes exactly zero difference. It is a cross reference included in both bookings, it does not entitle to through checking, and more importantly (especially outside of COVID flexibility) it does not result in protecting itinerary no2 if flight no1 is cancelled or late.

other airlines have (formally or informally) different policies on separate itineraries either in general, or within alliance, and typically linking itineraries makes no difference to those either.