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BA.com shows BA metal and BA codeshares on other airlines. It occasionally shows interlining with non codeshare airlines, but there aren't many examples. Essentially BA sells its own tickets online, in simplistic terms. Qatar offer a lot of BA codeshares from Doha.
It is possible that if ring the contact centre they will construct other fares for you, but if Qatar is offering a decent fare on their website and they are the main carrier then you may as well book via them, there must be hundreds of FTers who have done this without incident. For UuA, however, you must have a 125 ticket and preferably booked via BA. It is ok to UuA the LHR-DOH sector (presumably) and then to leave a DOH-Somewhere on BA flight numbers operated by Qatar. This complicated to do, and may require a long call, plus it's often the case that Qatar's own fares make a UuA less economically attractive anyway. However these are general comments which may or may not apply to your proposed travel.