Relations are indeed better between the mainland and Taiwan, however politically China still claims the island as part of the greater motherland. It would be sensitive for BA to operate to Taipei as it could be seen by Beijing as a political act, without going into great details as the the background it is not easy to explain the working of the Chinese mind in relation to Taiwan, just trust me it is not as straight forward as it may appear.
I know all too well how much of a mess this has become, this has been going on for over 60 years after all
What I don't get is why China still has a problem with Airlines in Europe (currently KLM only) in serving Taiwan under their own name (on the basis of the fact they are "symbols/arms of the state"), when they don't apply to American, Japanese, South East Asian, Korean and their own carriers (a fair number of which are state owned)
Remember the only nation that have diplomatic relations with the ROC in Europe is the Vatican, so I still don't get what Beijing has a problem with?