To amplify Yellow Pen's point I believe that mainland China wouldn't allow airlines of certain countries to serve both it and Taiwan. Hence the need to set up a separate airline to serve TPE, although its planes could go just about anywhere outside China and were doubtless operated by the staff of the parent airline. Apart from British Asia Airlines, Japan Airlines had a Japan Asia Airlines subsidiary and there may have been others. Mandarin airlines was set up as a partially owned subsidiary of China Airlines of Taiwan which did not carry the national flag making it more acceptable to the PRC (more details on wikipedia). For some reason the restriction never applied to American carriers.