The United Kingdom of Great Britain and Northern Ireland makes the notion of "country" confusing, since there are many different degrees of closeness between the constituent countries.
At times, the UK is considered a country. At other times, it is considered four countries (England, Scotland, Northern Ireland and Wales). Scotland and Northern Ireland have their own parliaments, but are ultimately subject to UK law. Somehow Wales is "more part of England" than the other two countries, and so does not have a big secessionist movement, or a parliament. It is also the only country in the UK not represented in the Union flag.
Sometimes the UK is referred to as "Britain" (probably because it was the seat of the British Empire), although that actually refers only to England, Scotland and Wales.
Sometimes, the UK is referred to as "England", much to the annoyance of those in Scotland and Northern Ireland. It may also annoy the Welsh, but they do not make a big noise about it. However, when the Scots or the Northern Irish wish to leave the UK, it is always because they wish to leave England, rather than each other.
The Queen of the UK is often referred to in other countries as "the Queen of England" (since that is the country of her primary residence), even though that is not one of her (many) titles. The Articles of Union tie the monarchies of England and Scotland. There are no articles of union tying these to Norther Ireland (because that union is too recent), or to Wales (because that union is too old). The Queen of the UK also happens to be the queen of 18 other realms (such as Canada, Australia and New Zealand), but these are not part of the same kingdom as the UK. For example, if a change in the rules of succession were made in the UK but not in one of the other countries, then there could be different monarchs, whereas the rules of succession for all countries in the UK are forced to be the same.
The "British Isles" refers to multiple countries, not all of which are part of the UK. In particular, it includes three Crown dependencies: the Isle of Man, Jersey and Gernsey.