My view:
China could not physically resist a British demand for and occupation of Taiwan, at least in the 2nd Opium War, which saw them march to Beijing and burn down the palace. They could well succeed in demanding Taiwan in the 1st opium war too. It's an island. Britain is good at conquering those, and at making sure enemy armies don't show up on them.
Once a British possession, Britain wouldn't sell it.
London could only use Taiwan as a *supplement* for Hong Kong and mainland treaty ports, not a substitute for them.
Uses for Britain of Taiwan: A naval, military and trade waypoint in the Far East. A base that can be food self-sufficient. From northern Taiwan it is a shorter hop to further up the Chinese coast, Korea, the Ryukyus and Japan.
Taiwan could be a multicultural environment, with South Asian coolies working there as well as Chinese and expat Brits. The size of the Indian population will depend on if the availability of India labor is higher and its price is lower than Chinese labor, at least to the point that it compensates for additional travel costs.
There could be a substantial body of ethnic Chinese troops recruited in Taiwan, who, like the Indians, could take part in imperial policing missions.
A British Taiwan is a Taiwan that is not available for the Japanese later. The effects could go further than that however. With a more robust and forwardly placed base in the Far East, Britain can exert more power there and possibly forestall power political rivalries between Japan, Korea and China. If Britain used its extra power projection to stop anhything like the Sino-Japanese war before it starts, or to head off annexation of Korea, the three countries may be less focused on each other as the enemy in the 20th century and get along. They could all see Britain as the big regional bully and agree on that.
That might make their geopolitical attitudes more like those of people in the Middle East, Africa and South Asia, who looking back would see Britain as the main villain. While there's no reason it would be more cohesive, there could be more genuine pan-Asianism in the region as one consequence of this.