Everything happens, or in this case does not happen, for a reason. Ultimately the British Empire is about money. How can Britain and it's citizens earn as much money as possible from resources overseas at as little cost as possible.
The section here about Taiwan under the Qing Dynasty probably explains why Britain didn't want it in 1895. Mainly an underdeveloped agrarian economy based on sugar, tea and rice. Britain already had sources for all these from elsewhere in the Empire.
Seems like a bit of a money pit, given what Japan had to invest in infrastucture and suppressing banditry from 1895 onwards. Piracy/banditry was a perennial problem in Chinese waters even into the post WW2 period. Part of the RN effort was devoted to suppressing it. Acquire an island as large as Taiwan and the problem, and associated cost, grows. Britain didn't need the headache. An expanding Japan was prepared to accept the cost.
Just 3 years later Britain was able to Lease the New Territories in Hong Kong so expanding its existing and very well established colony based on Hong Kong island and Kowloon peninsula. That had direct access to markets on the Chinese mainland. (Look up the Jardine Matheson company history as an example). And it had other interests on the Chinese mainland, in the Shanghai International Settlement (albeit that that was not itself a colony). It was also able to lease Weihaiwei as a naval base on the Chinese coast in northern China from 1898, after the Japanese had withdrawn, and at the same time the Russians leased Port Arthur. Being on the Chinese mainland itself offers far more opportunities to influence China.
1895 was also in the middle of the "Scramble for Africa" when all the European nations were seeking to carve out colonies in Africa to tap into the natural resources existing there and the relatively easy money to be derived from that. So for example in Southern Africa we have expansion of the existing British colony leading to the Second Boer War in 1899 as Britain sought to take over Boer colonies with their gold and diamond deposits.
So for Britain, there are far better economic opportunities in 1895 than taking up an offer of an additional and poor piece of Chinese real estate to help the Qing dynasty out of a hole.
But if you want to pursue the idea, the question becomes why, given its location, is it any more defendable than Hong Kong in 1941? How much does it delay the Japanese push south? Maybe they just restructure their plans a bit? Delay invading the Philippines by a few weeks/months? While Philippines defences might be a bit better prepared, the overall US/Philippine Defence Forces are not going to be hugely different.