Team GB enters the Olympics but as you may have noticed it was England who came Third in the World Cup recently, and likewise England play international rugby, including against Wales and Scotland.
† The exact rules are complicated, but they don't depend on where on the island you were born or stuff like that.
> Rugby in Ireland is governed by the Irish Rugby Football Union (IRFU), which was established in 1879, decades before Ireland was partitioned in 1921. It has always organised rugby on an all-Ireland basis.
Except when they play as part of the British & Irish Lions!
So in modern English they may be states, but there sure is a lot of overlap between "country" and "state".
As an English person I would say that Scotland and Wales are nations and countries; constituent countries of the country that is the UK.