There normally is a bus from T5 to T2 but since you're arriving from outside Schengen I don't believe you can use that. I'm also fairly confident that due to the change in terminals you will have to collect and recheck your luggage. It's about a 15 minute walk when you're landside between the two or you can take a buss.
Immigration at ARN is efficient and predictable if you are in the front of the plane and/or hold EU citizenship. You could theoretically re-enter T5 with a quick security check after immigration and take the T2 buss but it won't safe you any time.
Check-in with BA you should do with their app, getting airside is a <10 minute process once you are in T2. There are 5 flights a day with BA to LHR but if it's not a single itinerary they would not rebook you by default. The DOH flight arrives at 13:00 so you should get onto the 15:00 or 18:50 BA flights. If there are any issues with those there are plenty of SK flights to get rebooked to as well.
I wouldn't do this with at least a 1,5-2 hour layover.
Last edited by FlyingMoose; Jan 4, 2015 at 12:17 pm