Northumbrian

Obviously with separate British nations, we will see separate National Olympic Committees, so that takes care of the sports teams. And where there is government funding there will be no more British organisations either.
And politically, the Council of the Isles would be on a British/Irish basis.

But there may be some hope for the Brits. In North America, professional sports leagues exist for ice hockey, baseball, basketball, football (soccer) to include both American and Canadian teams (though why a country of 30 million people can't organise their own professional sport is beyond me).

But whereas now, a person wanting an English organisation to break away from a British one is regarded as a bit of crank (protocol is to wait for Scotland and Wales to break away first), the position will be reversed for somebody wanting to form a British organisation post-independence.

Any offers?