I did this connection in May, albeit coming off a SK flight, with a slightly longer window of 1h35 on unprotected tickets - which made me a bit nervous!
There are both airside and landside options - I've done both, and there's not much between them on a Schengen to non-Schengen transfer. As you're on non-Schengen to non-Schengen, I suspect the airside route will be quicker for you. What I'm not clear on is whether you'll get a security check somewhere though - I had a US-CPH-ARN itinerary, and so far as Schengen is concerned the US is a trusted country. I'm not sure if you'll find the same with Qatar.
The airside route for me involved passport control (which presumably you won't have), down into the basement area (signed as a gate, iirc), telephone for airside bus transfer, 10 minute wait, 5 minute bus ride direct to the BA gate, then up the stairs into the gate area.
I took the landside route last time, and I think I went from plane to lounge in well under 20 minutes, albeit security at T2 was completely empty.