The answer lies in the history of individual sports, mollykins. In soccer, there have been separate ruling bodies for the 'home countries' since they competed 'internationally'. England, Wales and Scotland have still got wholly disproportionate power within the ultimate governing body of soccer, as I recall, having a vote each in the six voting countries that decides on changes in the laws of the game !
In rugby union there wouldn't have been regular internationals if the countries weren't separate.France and Italy are comparative newcomers to the 'home' internationals.In rugby league there is a Great Britain team, however.
In athletics, Great Britain was always represented by athletes wearing the 3 hooped vest of the Amateur Athletic Association,which is an English based association.They wore that,if they were members, when not competing internationally.The four minute mile was broken, in Oxford, by a runner wearing that vest, and the others in the attempt wore it too.
So, you see, there's a mish-mash of reasons for the curious way in which national representation is as it is now..
And the Union flag ( known as the Union Jack) is the flag of the 'United Kingdom of Great Britain and Northern Ireland', as 'Great Britain' or 'the United Kingdom' is known for short, isn't it ? We're the 'UK' or United Kingdom to the United Nations and economists and Britain or Great Britain or 'GB' for most other people.The official title is too much of a mouthful for everyday use.