Remarkably, the dissolution of the British Empire has a specific date: the London Declaration of 28 April 1949, which created the voluntary Commonwealth. Over the following twenty years, every major British colony except for Hong Kong chose to become independent, reducing the former empire's overseas population to 5 million (from its peak of 700 million).
American military historian Chalmers Johnson credits this decision with preserving the UK's status as one of the world's wealthiest and most influential developed nations--in contrast with, for example, the Soviet Union, which continued to invest in an unaffordable empire up to the point of collapse.
To be honest that's a hard question to answer. There isn't a date which everyone in the UK said we are not the/are a world leading power any more.
To answer your question to the best of my knowledge (I might be wrong with this, so do please correct me) the warning shots were probably back in 1899. In 1899 Germany and the US (who were overtaking the UK in terms of industry) "split administration of the island of Samoa", without including the British Government.
Then you had the Easter uprising/War of Irish Independence. I can't find the quote at the moment but I am sure I read one that said something along the lines of: The British have willing given up core territories; this is a sign of a nation in turmoil.
You then had the Statute of Westminster which made the Dominions more like independent countries back in 1931 which would have marked in some eyes the start of the end. Later still, in 1947, you had the guarantee of Indian Independence. This lead Mr. Powell, who wanted to be Viceroy of India, to walk the streets of London to come to the conclusion that if India goes so should the empire.
I think what really drove the point home for the public; that the British Empire wasn't in charged any more was the years after WWII. It was US money, US military, and US culture which protected and grew the UK against the USSR. Countries were forcing their independence, becoming republics.
We already have plenty of excellent answers detailing when Britain (in practical terms) stopped being the worlds leading power, as for when the public realised that is more complex. However one series of events destroyed the confidence of the British public in the power of their nation more than any other - the Suez crisis.
The Suez crisis did massive damage to the conviction of the British people, as not only did the attempt to retake the canal ultimately fail, making Britain appear militarily weak, but the British people lost confidence in their own leadership, leading to the resignation of the Prime Minister Anthony Eden.
The Suez crisis also showed that Britain could no longer act with impunity on the world stage, and would require the consent if not assistance of the USA in her future military endeavours on foreign soil. A sure fire sign that a nation is ceasing to be a world power is it's reliance on other nations and inability to act independently without fear of major reprimands. The Suez crisis was in many ways the final nail in the coffin of British imperialism, and would have a profound impact on Britain's foreign policy in the coming decades (all future leaders would seek sweet relations with the USA at all costs [especially Harold Macmillan who succeeded Eden.]) So through the turmoil it caused at home and abroad the Suez crisis proved to the British people that Britannia was no longer the worlds leading power.
As /u/king_dumb points out, it's hard to pinpoint a specific date for this sort of thing. However, I reckon the first glimmers of doubt began to emerge as early as the 1870s. The idea that America would inevitably usurp Britain's position on the world stage was certainly in wide circulation by 1900. The following passage comes from my PhD thesis. It's the introduction to chapter 2:
In September 1878, William Ewart Gladstone made a prediction. “The England and America of the present”, he argued, “are probably the two strongest nations of the world. But there can hardly be a doubt as between the America and the England of the future, that the daughter… will [soon become] unquestionably… stronger than the mother.” The United States, he warned, was “passing [Britain] by in a canter” and would soon usurp its position on the world stage. It was essential, he concluded, that his countrymen recognise the inevitability of this impending transition and begin to address what he later termed the “paramount question of the American future.” The publication of this prophecy is important, for it marks the onset of a transitional period in Anglo-American relations. Over the next two decades, the idea of America in British culture underwent a profound transformation – one which altered the transatlantic balance of power and had significant ramifications for the reformation of British national identity. At the heart of this process was a shift in Victorian conceptions of what this chapter terms the ‘American future’. At the start of the period, the response of Victorian journalists to Gladstone’s warning was almost universally hostile. The Times concluded that his reasoning was “at once redundant and defective” and was intended only to massage the ego of its American audience. The Morning Post accused Gladstone of seeking some form of revenge against the country that had rejected his party at the ballot box. Provincial papers, such as the Blackburn Standard, roundly condemned the ex-premier’s “unpatriotic”, “wild”, and “sinister” prophesies. Even the Liberal-leaning Daily News accused Gladstone of having “gone too far” with his “rash”, “sensational”, and “astounding” depiction of America’s “unrivalled future”.
It is possible to detect an undercurrent of anxiety running beneath these protestations of national confidence. Whilst The Graphic was quick to reassure its readers that “the facts [do not] justify Mr. Gladstone in his prophecies”, its defence of Britain’s international pre-eminence was less convincing. “The Americans have far more land than we”, it began:
but their intellectual, moral, and physical forces can never be so well organised… Besides, England is the centre of an enormous Empire... Even if there were some doubt on the matter – which we do not admit that there is – it could hardly be wise for a statesman… to address such flattering words to the Americas.”
The necessity of affirming Britain’s superiority, rather than simply taking it as read, is of course the most telling feature of this collective response. The fact that the press responded to Gladstone’s comments in such a defensive manner suggests that the threat posed by American competition was felt more seriously than journalists were willing to admit. However, whether they were fuelled by bravado or genuine confidence, the fact that most papers refused to entertain the possibility of Gladstone’s hypothesis is significant. It suggests that the idea of an inevitable ‘American future’ in which Britain would play a subordinate role to America was, at this point, regarded as culturally unspeakable. Indeed, the idea was so toxic that Gladstone’s allies in the Liberal party – men who were aware of America’s rapid economic progress – immediately distanced themselves from their former leader and publicly countered his prophesy with optimistic predictions about Britain’s enduring prosperity.
Twenty years later, when the country mourned Gladstone’s death, this cultural landscape had changed. Whilst some British observers continued to deny, or at least resist, the United States’ growing influence, the balance of public opinion had tipped decisively. The concept of an inevitable American future was now in widespread circulation. By the end of the Victorian period, it was commonplace for commentators such as F. A. McKenzie to claim that an ‘invasion’ of American products and inventions had swept the country, for the Daily News to joke that British children should be taught American English (‘the language of the future’), or for W. T. Stead to describe the Americanisation of the World as the “trend of the twentieth century.” No longer regarded as an unstable political experiment, or dismissed as an underdeveloped post-colonial backwater, the United States was increasingly portrayed as a land of economic and technological progress, an influential player on the international stage, and the home of a distinctive brand of social, cultural, and spatial modernity.....