It seems to be that BA is particularly fussy about accepting bags and interlining them on, I recently asked about interlining bags CX-BA-LH (i.e. several OW sectors then onto *A, the final *A sector on a totally separate booking). CX told me they were more happy to do this, although it turned out they couldn't due to too many sectors.
So, this is basically up to FlyDubai, not BA, and according to
http://www.flydubai.com/en/news/arti...light-to-juba/
The airline has also signed a number of interline agreements with carriers including Emirates, Singapore Airlines and British Airways to ship goods beyond flydubai’s passenger network, which now spans more than 55 destinations.
You might want to check directly with them, though...