Not totally relevant to BA, but I assume it will be the same process. I booked a ticket last week with LY from JTR-ATH-TLV with the first flight on Sky Express. The check-in agent managed to send my luggage all the way through to TLV but couldn't print my boarding pass. In ATH, I went to the LY check-in counters for the regular questions and they asked to see my baggage receipt for the bag sent to TLV. After they gave my passport the usual sticker and labelled my hand luggage with the same coloured sticker, I managed to collect my BP. Upon arriving in TLV, I noticed my bag had actually been given a new tag from LY.
So it could be that even connecting passengers need to exit the airport and go to the LY check-in desks for a security check and then to get their boarding passes.