If Britain tells America to fuck themselves, the British Empire still collapses and dissolves because America isn't what killed it.
When Ireland left, the Empire entered a terminal decline, which was hastened by India's exit. A wider phase of decolonisation followed these two events, with the Commonwealth of Nations formed to facilitate decolonisation but also keep the newly independent countries close to the UK. America's rise as a world power, whilst a contributing factor (replacing Britain as the major economic and industrial power following the First World War when, ironically, the British Empire reached its furthest extent), Britain didn't "bow down", but instead the economic and political damage of the First and Second World Wars saw the rise of successful independence movements as Britain's prestige, influence, and confidence on the world stage took a metaphorical axe blow.
That said, had Britain told America to shove off, Britain would have entered the EEC sooner (no close relationship with America means that de Gaulle won't see Britain as the backdoor for America to get in), which only means that the Empire dissolves quicker as Britain moves into a closer bond with Europe and is less reliant on the wider Commonwealth.