Yes, CI flew in/out of HND up until a couple of years ago when they moved to NRT. Actually, the reason why CI was forced to fly into Haneda back in the late 1970s (when all other international flights were moved to Narita) wasn't necessarily because Japan switched diplomatic recognition to the PRC, but because at the time China's air carrier, CAAC (now Air China) did not want to fly into the same airport as CI. As time went on, this was kind of a silly reason, given that PRC and Taiwan (ROC) airlines flew into common airports such as HKG and BKK. And after getting over the initial humiliation of being forced to stay at the old, beat up Haneda airport (then), CI turned their lone representation at Haneda into a big selling point---it's a lot more convenient to central Tokyo than NRT. CI had very high load factors on these flights, not surprisingly.

Also, after awhile, the Japanese government WANTED CI to move to NRT because of the extra costs of having to maintain customs and immigrations facilities just for them at HND, and for many years, CI refused, because they were enjoying a good run being at HND.

Japan Asia Airways (JAA) was another child of the political contentiousness between the PRC and Taiwan, it was set up specifically for Japan-Taiwan flights after Japan switched diplomatic relations to the PRC and JAL stopped flying to Taiwan in the mid 1970s. I don't think it flew out of Haneda after international flights were switched over to Narita in 1978, though. They've had a reputation for flying cast-off, older aircraft from JAL and some Taiwanese have interpreted it as a snub to Taiwan for that.

BA and AF have also set up similar units, called British Asia Airways and Air France Asie, respectively, though I think it's just a formality and they fly the same planes going on to Europe.