> which for example each compete in football

Depends on cont context. In the Olympics there is a UK football team.

They compete as team GB, further demonstrating the madness of trying to define a nation

Team GB is a trademarked brand name for the United Kingdom olympic team. (I think because "GBR" is the established letter code used so GB has some brand recognition factor in an olympic context.) It's controversial since "GB" can be seen to exclude NI although the team includes the whole UK.

It's even more than UK. Includes different Crown Dependencies like Isle of Man, British Ovsersea territories like Montserrat or Gibraltar. https://en.m.wikipedia.org/wiki/British_Olympic_Association

With the British it's always complicated :)

The four parts of the UK are called “countries” for purely historical reasons. There’s no actual concrete difference in political organization today that would let one claim that Scotland is a meaningfully different type of entity from autonomous regions of other unitary states (e.g. Catalonia).

UK isn't really made of countries in any internationally recognised sense. The "four nations" are really are just administrative regions with a trumped up sense of identity. Having different football and rugby teams is a a historical accident since those sports were invented here.

How is that different from nation-states? It's an accident of history that some Karelians became Finns while others didn't. It's an accident that Finns and Estonians exist as separate nations. It's an accident that Savonians and Tavastians became Finns but Kvens and Tornedalians didn't. It's an accident that Izhorians, Livonians, Votes, Ludians, and Vepsians were assimilated by non-Finnic peoples. It's an accident that Finns exist as a nation instead of becoming Swedes or Russians.

Administrative borders shape how national identities evolve and endure. And when long-distance communication and travel became easier in the 1800s, administrative borders shaped the new wider national identities.

I agree that accidents of history led us to where we are now. The four parts of the UK are regarded as nations in a loose sense, but not as nation-states.

The same way the rest of the world most of the time considers the US as a whole -- yet this does not make the individual states disappear.

Exactly. The "countries" of the UK are in some ways more like the US states. Each very much real with their own identity, traditions, systems of devolved government etc. Just not countries.