Sigh... No offence Lucky, but ocassionally you get so focused on the airlines you like (e.g. British Airways) that you totally forget that there are other airlines and other markets out there.
If you look at this article, you've totally missed out that IAG owns several airlines. Yes, your precious BA is one of them. But the other big one, which I believe is the main reason Qatar has an interest in IAG, is Iberia.
Qatar has a long history of trying to get into the spanish market, they looked loosely at Air Madrid, and more intensely att Spanair, opting out at the literal very last minute before Spanair went belly up. Now they're finally there, since IAG has their main office, registry etc in Spain, and not in the Uk.
If Qatar is looking for a market where they can grow and even have a hugely profitable fifth freedom flights base, it's Spain, not Uk. Spain has very few direct flights to north america and instead they are used to feed the large hubs in London, Paris, etc. Unreal considering it's a country with 46 million inhabitants and an economy that largely depends on tourism.
I could be totally wrong of course but I wouldn't be surprised if all these potential direct routes from Spain to north and south america is what Qatar is really after.