The United Kingdom (of Great Britain and Ireland/Northern Ireland) became a unified political entity as of the 1701 Act of Settlement, and the since that point the country was known as Great Britain ( abbreviated to the UK or Britain). During the age of colonisation Britain established a global colonial empire, and in 1876 Queen Victoria was proclaimed Empress of India, though she was still monarch of Great Britain and Ireland. Since 1701 the UK parliament was dominated by English landowners and politicians, with little Scottish, Irish, or Welsh say in the overall strategy of the empire. While Welsh/Irish/Scottish politicians such as Lloyd George and Ramsay MacDonald had a hand on the tiller at crucial points in British colonial history, they did so under a unified 'British' mandate, not a parochial Welsh/Scots/Irish mandate.

The British people still referred to their homeland as Britain, Great Britain, or the United Kingdom, while also celebrating the fact that they had an empire, which was globally known as the British Empire. Unlike the Empire of France or the German Reich, Britain never formally declared herself an 'Empire' in this sense, so while the monarch of was the King/Queen of the United Kingdom of Great Britain and Ireland was the Empress/Emperor of India, they were never called the Empress/Emperor of the United Kingdom et al.

There was also the fact that the age of Empire, in terms of celebration and public recognition, only occurred from around 1870 to 1950, at which point the de-colonialisation process was underway (India had gained independence in 1949). Britain then transitioned the empire into a Commonwealth of Nations, hence the British Commonwealth, with the British monarch as head of many of those nations (such as Australia, New Zealand and Canada). Once the Commonwealth came into being the notion of Empire slipped into history, and while Britain retained colonies up until the late 1970's (it still retains small islands in the Atlantic and Pacific), the notion of an 'empire' slipped away gradually. The decolonialisation process happened gradually, and as such the evolution of empire into Commonwealth enabled the British people to see the Commonwealth as the natural successor to the former empire, while also allowing the British to 'gracefully' exit their colonies without fighting rearguard colonial actions (though Malaya, Kenya, and Rhodesia had armed conflicts in the run-up to and after decolonialisation).

If you are not a British person, you also have to remember that the national identities of the individual components of the United Kingdom still have their own regional/national identities. The Northern Irish, Welsh, Scottish, Cornish, and English all have their distinct national identity, and while it is not unusual for a British person to view themselves as 'British', they are also likely to view themselves as Welsh, Scottish, Irish, or even Yorkshire before they say they are British. Britain has a complex social identity, and in the post Empire period these differences have become more marked, especially in de-industrialised areas such as the Scottish lowlands, southern Wales and Northern Ireland. The unity that Empire brought has been swapped for a more parochial vision, which has become increasingly more apparent in the English, which is why the Brexit vote was such a significant one, as it pitted the Scots, Northern Irish, and European facing English against those parts of England that had suffered under the neo-libral ideas of Thatcher and Tony Blair's governments. The revanchism, and appeal to a golden age of 'England' (namely a white dominated, Johny Foreigner excluding Empirer), resonated with voters who were desperate to recapture a perceived golden era that the late British Empire conjured up in their minds. (Note to the mods: I appreciate that this is under the 25 year rule, but it helps bring context to the question).

Ultimately your questions are complex, namely in that while the diplomatic nature of the UK never acknowledged the Empire (vis-a-vie France, Germany, or Russia) in its title, globally the sobriquet for the UK was the British Empire. It was not unusual in this respect, as the USA look this approach when it established colonies after the Spanish/American war. In terms of when did the British stop seeing themselves as an empire, it depends on what generation you belong to. The post-Commonwealth generation have no emotional ties to the notion of Empire, and the generation that grew up under the Empire appreciate that it no longer exists, though there is an undercurrent amongst the English in particular that hankers after the role the United Kingdom (under English rule) played in the world during the colonial period.