The British Isles [a geographical not political location] is composed primarily of two islands, Great Britain and Ireland. Great Britain is divided into three, England, Scotland and Wales. Ireland is divided into two, Northern Ireland and Eire.
As far as countries are concerned there are really two. One is the United Kingdom which contains all of Great Britain and Northern Ireland. The other is Eire. Many sports and other organisations do not follow this distinction.
In Bridge at European and World Championship level there are two countries, Great Britain and Ireland. The British Bridge League [BBL] sends teams to such Championships, as does the Irish Bridge Union [IBU]. Most competitions and events are run by the bodies representing each smaller country, ie the English Bridge Union [EBU], Scottish Bridge Union [SBU], Welsh Bridge Union [WBU], Northern Irish Bridge Union [NIBU] and Contract Bridge Association of All Ireland [CBAI]. In fact the BBL runs a few events itself for funding purposes.
The EBU, SBU and WBU are members of the BBL and part finance it: similarly the NIBU and CBAI are members of the IBU. There are major events between these countries, the most important being the Camrose between EBU, SBU, WBU and NIBU. After twenty years or so of talking, the CBAI has just joined this series.