BA definitely will not facilitate checking you through to QR. You can ask but don't expect them to agree, they have a written policy of not doing this since around 2017.

And I also think you need to go airside for BA to QR at OSL. This question of shuffling between BA and QR has been asked a few times over the years, and unless there has been a chance BA to QR you definitely have to go airside, and also have to walk the length of the entire terminal as the airlines operate at different ends of the terminal.