i did QR to BA on Monday. QR lands at T5, and being one of the first out but landing behind Emirates and a UK flight it was about 10 minutes from doors opening exactly on time until I made it through passport control, then at another 30 minutes to get my bag back. At that point, your flight took off five minutes ago.
Once I had my bag, and dropping it off at business check-in with no queue, walking quickly it took me twenty minutes to transfer between terminals and get through priority security. So, with all that in mind, and allowing for the delay on my bag, I don't think anything less than an hour is physically possible. IF you are hand baggage only and there is a slight delay on your BA flight, there is a glimmer of hope that you'd make the BA flight, but the odds are basically negligible.