The way these islands divide on sporting terms is idiosyncratic to put it mildly.
Olympic Games - Uk and the Irish Republic. This is political and legal reality as recognised by the rest of the world for the last 100 years.
Football - England, Scotland, Wales, NI, Ireland. One independent state and four top level sub-divisions. Other countries top level subdivisions do not get their own national teams. Lombardy does not play Catalonia. Texas does not play Sichuan.
Rugby - England, Scotland, Wales and Ireland. This reflects the political reality as it last was in 1284.
Commonwealth Games - England, Scotland, Wales, NI, IoM, Jersey, Guernsey. Fuck knows.