British Asia Airways was formed by BA
to serve Taiwan. The aircraft were essentially standard BA
apart from the changed titles and two Chinese characters (meaning British Asia) on the tail with the Lander aircraft. When the World Tails were launched BA
had the aircraft carry the Chelsea Rose and Benyhone Tartan schemes. Two aircraft were in the fleet at any time, so that one was always available to serve TPE
. British Asia Airways was wound up when BA
dropped its flights to TPE
(Which operated as BA25 outbound and BA26 return).
As well as Air France and KLM having similar Air France Asie/KLM Asia subsidiaries, as mentioned, Swissair also had a Swissair Asia subsidiary. Qantas served Taipei with its two 747SPs which operated under the Australia Asia Airlines banner, while JAL has JAA (Japan Asia Airways) to serve Taiwan.
The reason for such subsidiaries existing was due to the sensitivity of the Mainland Chinese who regard Taiwan as a renegade province. I'm unsure why only some carriers who served China and Taiwan had to have separate subsidiary carriers to serve the latter, but think it is related to whether their home nation recognises Taiwan as an independent state.