There's a lot of strange information being thrown about in this thread.
You CAN check through bags for flights made on one booking/ticket.
You CAN still book an Avios (airmiles) tickets with connections, subject to availability obviously.
BA policy is NOT to check through bags on separately purchased/ticketed reservations. So if you purchased a ticket from EDI-LHR and then had a separately purchased ticket to LAX they won't (or shouldn't) check the bag through. This has been BA policy for a while now.