BA is steadily pulling out of the Qatar market so most of the flights sold under BA are now flown by Qatar. Note that there is no dedicated BA desk at Hamad Airport. This is relevant if something goes wrong with the ticket. I can give you two examples:

1. I booked a multi-leg flight originating in Doha on BA, first leg was Doha to Edinburgh, flown by QA. At check in online I was redirected to Qatar Airways website to proceed with the check in and in the process something happened to my ticket name and it became corrupted. I could not check in. The airport refused to help me as the ticket was originally a BA ticket, not a QA ticket. I had to buy a separate one-way ticket to Edinburgh on QA to salvage my holiday. At Edinburgh BA reviewed the ticket and to make a long story short, two months later I finally received acknowledgement that something had gone wrong in the computer systems in the redirecting from BA's website to Qatar's website and that changed my ticket name (first name was changed from a commonly recognised first name to pure gibberish) and I got the refund for the second ticket.

2. A coworker had a ticket on BA but flown by QA for his flight to the UK from Doha for Christmas. Arrived at Hamad Airport only to discover that the flight had been cancelled. No notice was sent to either him (his email address was attached to the ticket) nor to the corporate travel office that booked the ticket for him.

In both cases because there was no BA desk at Hamad Airport, there was no one to talk to and the BA phone numbers were useless or unreachable, as you can imagine.

I've heard other confusing stories about the BA-QA codeshares and while I'm sure it's smooth and fine for most passengers there are still too many anecdotal disasters so I'm now highly reluctant to book a BA flight that is flown by QA and requires checking into QA separately. If the cost differential is only marginal, book directly with QA for peace of mind.

I've largely stopped flying BA because I want to build up my mileage on Qatar. Qatar's reward miles are far easier and more generous than BA's reward miles via Avios, which is nearly impossible or pointless to use if you only want to book reward flights. After a handful of QA flights I now have enough QA reward points to fly round trip to Europe without any additional costs, while I have 120,000 BA avios points that are virtually worthless.