This is my idea how this happen. Japan during the Meiji Era started the modernization of Japan and started copying everything western. I believe even the sailor suit used by school children was copied from Britain. As for the bell chime, it seems it was started only near 1950s.
According to this site (In Japanese), after WW2 around the 1950s, the school chime at Tokyo, Ota Ward State Oomori 4th Junior High School (東京都大田区立大森第四中学)'s bell used for indicating start and end of class was always requiring repair, they started looking for a new chime. There are already lots of manufacturer that have Music Box (オルゴール) that contains the well known Westminister『ウェストミンスターの鐘』 melody in them. After that chime was used for elementary school nationwide.
Here is the Japansese reference copy.
Note: In Japan the Westminster Quarter is also called by the onomatopoeia word Chin-Con-Kan-Con『キーンコーンカーンコーン』, which is literally what it sounds. That made it difficult to search from the net.
Then South Korea and Taiwan were colony of Japan for 50 years although they were independent country after WW2, I believe since Japan established most of the school institution in the two countries, and given that Japan was the powerhouse economy in Asia after WW2. There is a big chance they copied the chime bell used in school. Still need reference for this.
As for why South East Asian Chinese school also have the same chimes, before the resurgence/opening of china (before mid 90s) most Chinese school in south east Asia follow Taiwan as a reference in their Chinese education. Chinese South East asian children studying in Chinese school would send their children to Taiwan to learn the language and culture. Schools would use Traditional Chinese instead of the now standard Simplified Chinese. I would assume that during these times, they would copy the Westminster Quarters as school chimes from Taiwan during those times and it stuck.
Britain -> Japan -> Korea and Taiwan -> Southeast Asian Chinese school