The past sovereignty of Scotland, England and Wales would be moot as they are no longer independent nor sovereign - if the requirement for participation in international RL or other international sports was that national teams represented sovereign states. It isn't.
The Cook Islands team represents the self governing (unlike England) parliamentary democracy that is
Cook Islands, which it has been since 1965. It is in
free association with NZ, and the Cook Islands has foreign relations with a host of other nations across the world.
Foreign relations of the Cook Islands
Incorrect, both that the Olympics only requires sovereign states to participate, and that it is the only exception (to what exactly, I'm not sure).
National Olympic Committee
The USA and Puerto Rico can play each other in basketball, and both participate in Olympics, and a Puerto Rican could be eligible for both.
A Northern Irishman could be eligible both for the Republic of Ireland team, and Team Great Britain. In soccer, Northern Ireland (NI) field a national team in soccer when the Republic of Ireland(ROI) team can select anyone off the island of Ireland, as Northerners automatically have ROI citizenship under the ROI's constitution. The right of ROI to pick NI players has been ratified by the International Court of Arbitration for Sport. The reason Northern Ireland field a team is because some of its people want a team to uniquely represent NI. This does not mean the ROI represents NI, it means the ROI team represents the ROI, of whom Northern Irelanders are citizens.
Similarly, the Cook Islands RL national team represents the Cook Islands only. The NZ national team represents the state of New Zealand, not the Realm of NZ. Cook Islanders happen to be eligible for the Kiwis because Cook Islanders are NZ citizens - however, the Kiwis still represent the state of NZ and not the Cook Islands.
Also, in soccer, both the American Samoan football team and Team USA could qualify for the FIFA World Cup. So, an American Samoan could be eligible for both teams. However, the USA team only represents the USA, and American Samoa represents the territory of American Samoa.
If a Taiwan team and a People's Republic of China team wished to participate in RL, would you object to the Taiwan team on the basis that the PRC claim that Taiwan is a part of One China?
Your classification of the NZ Cook Islands relationship in the first instance was incorrect, so that renders this comparison flawed.
