Regarding Phuket: So if on one 1 ticket, yes.
If on 2 tickets ie:
HKT>DOH>CPH on QR on ticket 1,
Then CPH to LHR on BA on ticket 2,
then QR's own policy and the OneWorld policy applies. The OneWorld policy doesn't define "connecting" flights, which in a transport for carriage contract used in the various airline conventions, means they have to be on one ticket. The OneWorld policy doesn't state that it uses such a defined term, so YMMV. QR have no issued policy for themselves. Anecdotally QR appear to be less inclined to check through to OneWorld if on separate tickets. OneWorld don't/can't police member compliance with policies, it seems to just be a marketing tool with frequent flyer mile cross airline allocation.
So if you are on 1 ticket, yes it should be checked through. If on 2, OneWorld policy is helpful but not precise in its definition of 'connecting', so you can but ask at HKT. If declined, show the OneWorld policy, if further declined, not much you can do. BA to QR on the outbound is nice (and correct) but has no standing for QR to BA on the return.
Hope you get a good result.