From the outside. On the inside, I think most Britons identify more as English, Scottish, or Welsh than British. Scotland and Wales live under the semi-official delusion that they are actually "countries" in some meaningful way, not just autonomous subdivisions of the UK, and even compete separately in international sporting events outside of the Olympics. Even forming a unified soccer team for the Olympics that the UK itself is hosting was a source of significant controversy.
The controversy mostly stemmed from the potential effects of running a British football team, particularly regarding the UK's status within FIFA.
The home nations each have a representative within FIFA, UEFA and IFAB, as well as being able to compete individually, and collectively enter more club teams into the UEFA Champions League. This causes some tension with other countries, who see the UK as a single country that wields undue influence in the global game.
If the UK establishes its own football team for the purposes of the Olympics, then there is a fear that FIFA would use this as an excuse to shut down the Scottish, Welsh and Northern Irish FAs and force them to fold into the FA. This is the reason Scotland refused to allow its players to be picked for the Olympics.
The Scottish FA never had the power to "refuse to allow" anything, there just weren't any Scottish players worth calling up. The Welsh FA made the same statements but that didn't stop Welsh players from being called up.
Britain's status in FIFA demonstrates just how poorly the notion of a "British" national identity has taken hold.
From the outside. On the inside, I think most Britons identify more as English, Scottish, or Welsh than British. Scotland and Wales live under the semi-official delusion that they are actually "countries" in some meaningful way, not just autonomous subdivisions of the UK, and even compete separately in international sporting events outside of the Olympics. Even forming a unified soccer team for the Olympics that the UK itself is hosting was a source of significant controversy.