actually, when Japan colonized Korea, it changed the name from 汉城 (Han city) to 京城 (Capital city), because Japanese though 汉 was "Chinese/China/Han ethnicity" to avoid mixing.It actually worked, few years ago South Korea demanded China and Japan to change the name of their capital Seoul in Chinese/Japanese language from 汉城 to 首尔. As far as I known lots of Chinese medias complied..
The act is stupid by all standards of course. 汉城 was not a name imposed by Chinese, but was Korean's own choice when they named the city using Hanja. It was then adopted by China and Japan due to the fact that Hanja is the same thing as Chinese character and Japanese Kanji.
India (according to wiki) is ultimately a word from Sanskrit which is Inidan culture. It then was taken by Persians and passed on to the rest of the world. It has nothing to do with British colonial rule. Essentially India is similar to China (Qin) passed on by Persions to the west.
So Korea just carried that concept forward, Seoul 서울 means "Capital city" in native Korean. They didn't want to use the Hanja 京 (Capital) to repeat Japanese decision so they opted for the native pronunciation which had no Hanja equivalent... Talk about nationalism.... it's really the Japanese fault for instilling 汉 as "China/Chinese" when 汉城 means "Great city" in Korean Hanja for millenias.