So, there're the British Isles, The United Kingdom, and a separation, I gather, of Ireland from Wales, England, Scotland
It's even slightly more complicated than that, because the "United Kingdom" consists of England, Scotland, Wales and Northern Ireland. It is the Republic of Ireland (South) that is excluded from the "United Kingdom".
I am not even sure about the status of the smaller islands such as the Isle of Man and the Channel Islands in relation to the United Kingdom, let alone foreign possessions (or whatever you call them) such as Gibraltar and the Falklands.
A more useful distinction that people often get wrong is between Britain the island and the British Isles.
Britain - is the geographic island that consists of Scotland, England and Wales.
British Isles - is the archipelago that includes Britain, Ireland, the Isle of Man, Orkney, Shetlands, Isle of Wight, and other small islands between the Atlantic and the North Sea.
So Dublin, Belfast, Lerwick and Douglas are in the British Isles, but they are not on Britain, whereas London, Edinburgh and Cardiff are on both.
Apparently there are 194 inhabitied islands in the British Isles. Britain is the largest but there are many more. Here is a list:
https://en.m.wikipedia.org/wiki/List_of_islands_of_the_British_Isles