In a comment the OP makes a point that if Wales win the world cup (unlikely in (association) football, but possible in rugby) then it is only Wales winning, and not the UK.
This is the point. If the UK were to enter a unified team, then it would be the UK winning and not Wales. The national identity in Wales and Scotland is tied to Wales and Scotland, and not to the UK. A Scot (perhaps the proverbial "True Scotsman") doesn't cheer for England, and doesn't cheer a UK team of 10 Englishmen and Gareth Bale. "Anyone but England" as the refrain goes.
The English would dominate the team. The English football fans would see marginal improvement in the chance of winning, but a win would be less satisfying, polluted by the presence of "foreign" players.
For political reasons England, Scotland, Wales, can't send separate teams to the Olympics. The IOC invites only UK representatives. But there is a long tradition of the football teams being permitted separate entries. Indeed, until 2012 the UK didn't enter the football competition at the Olympics, as there was concern that fans would complain, and the separate entries to the FIFA world cup might be lost.
I'm not sure that there is any analogy in other sports, or other countries. Football was invented at English Schools, codified in London, spread to Scotland and Wales, before going on to conquer the world. There are few countries that have the particular structure of the UK: Separate cultures, laws, education, and politically distinct but united. Would the United States welcome the opportunity to field a "North American" team with players from Mexico, Canada, even though probably few actual USAians would make the squad? That team would be very strong, would have a very high chance of winning, but would the fans in the USA actually cheer a team composed of 9 Mexicans and 2 Americans? Would Mexico feel that their wins were devalued by the presence of Americans?
The desire of English football fans is for En-ger-land to win, not Scotland, and not a mixed team. England and Scotland may be parts of the same country, but in football, we are rivals.