BA will definitely not, but that's the outbound which you aren't asking about (and where I would always stay overnight or have at least 2 "rescue" flights available if same-day)
QR will usually be able to tag your bags to DOH-OSL-LHR on separate tickets, but to be sure you should double check on the QR forum whether your destination airport's QR check-in agents are funny about it.
I'm not too bothered about through checked baggage as I was planning on the assumption it wouldn't happen but it's good to know it's worth asking sbout, thank you.
The outbound LON-OSL is being planned to have a weekend in Oslo so definitely have no need for through checked bags. I don't mind being late home but not taking any chances on missing my flights out!