There were rumours that George III was having some sort of health problems, and just used his dislike for the government as an excuse to retire early rather than him, and by extension, Britain, being perceived by France and Prussia as being weak. Personally, I'd call his bluff on that, as he died at a very old age of 91 in 1830 during the reign of his great-grandson, George V.
Aside from future kings and queens abdicating or threatening to abdicate whenever they dislike their governments, there may also be effects from the very fact that George III decided to 'retire' in Hanover. The Prussians, who wanted to stop the revolution going on in France, was inexplicably denied transit through Hanover by George III, who was still sovereign of Hanover at that time (and for the next fifteen years after that). Would George III had the same audacity and fortitude to demand such a thing had he remained in London?
And when Director Bernadotte's forces tried to embark on a punitive expedition against the Danes 12 years later in 1805, they were also inexplicably denied transit by George (who by then had also abdicated the Hanoverian throne and was acting as Viceroy of Hanover for his son, George IV), citing that his 'flying baskets' can see and shoot at anything. What would have happened had Bernadotte made his way to Denmark? Imagine Denmark being ruled by a Bernadotte dynasty!