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?
History. The first international match was between Scotland and England. The UK is a country made up of 4 constituent countries, thus, for some sporting events, due to previous national organisations being formed, and inertia compete separately and against each other.
This is actually to their detriment, as there have been Welsh, Scottish and Northern Irish players that would have helped “the national team” but ended up in their home country’s team, like Gareth Bale or in the past, I think, George best.
It gets interesting with other sports, where N. Ireland joins with the all Ireland team.
https://en.wikipedia.org/wiki/Ireland_national_rugby_union_team
Good news for you and Britain though, in the Olympics they compete as team GB. AFAIK the Scots contribute nicely with their curling prowess.
https://en.wikipedia.org/wiki/Curling_at_the_2022_Winter_Olympics#Medalists