Wales, Scotland, and England have their own teams (Northern Ireland contributes to the Irish team). The common explanation I’ve seen is that the nations have deep, distinct histories and cultures, but that doesn’t explain why, for example, India doesn’t get to send a team from each state despite India’s ethnic groups being even more diverse and distinct than those of the UK.
So what is the actual reason for Britain into three parts for the purpose of international sport but not other countries?
They actually don’t, NI has their own side and they have qualified for the WC before (1958, 1982 and 1986).
The reason for individual teams is partly traditional/historical and partly because the UK is a union of nations, with the nations each having their own footballing representation for well over 100 years. I don’t know the specifics of India, but it’s not (as far as I know) a union of different pre-existing nations.
Also, given that despite their huge population India have never qualified for the World Cup, it seems unlikely that individual states would do any better.