This source draws from newspaper articles of the time. It dissects differing viewpoints among journalists in the UK regarding the American War of Independence. Many journalists saw the American War of Independence as unjustifiable. A rebellion initiated by a colonial minority who adopted extreme narratives depicting British and American relations as "slave owner and slave" in order to further their agenda for American independence.
Historian Historian Stanley Weintraub's "Iron Tears: America's Battle for Freedom, Britain's Quagmire, 1775-1783." describes British civilian sentiments towards the American Revolution as a feeling of betrayal. In this book, he described how Britons felt that the American colonists owed them a great deal for protection, for purveying their culture, for providing them with manufacturers.
https://www.npr.org/templates/story/story.php?storyId=4727956
However, King George III's views, expressed in an essay he wrote after the Treaty of Paris (1783) were that of a different nature. There's some speculation that his views were very influenced by an agricultural theorist, Arthur Young. In the essay he describes how Britain may actually gain more from having America as a trading partner as opposed to a colony. He was likely correct with that assertion, as Britain was now free of defense costs (which were quite substantial) but could still make considerable sums of money in trade. Britain also retained Newfoundland (at that time the largest fishery in the world), Caribbean possessions giving them an advantage in the sugar trade, and Rupert's Land which would ensure British dominance in the fur trade.
http://www.historywiz.com/primarysources/george3rd-letter.htm
https://www.royalcollection.org.uk/collection/georgian-papers-programme/america-is-lost
As for what the commoners thought? We can't really be sure. There weren't any opinion polls back then about the war, and there certainly wasn't very much emphasis in the press about what the common people thought about the war. Judging from Arthur Young's work, King George's essay, and the unusually favourable terms the Americans received at the Treaty of Paris in 1783, I can think it can be deduced that the British primarily saw the American colonies as a business venture. The goal of the empire was to make money, not necessarily to spread British civilization. An anglo speaking nation who traded with Britain was just as good... maybe even better... than colonies that were expensive to defend, and full of what many Britons viewed as ungrateful inhabitants.
This sentiment would also explain the negotiations of the Treaty of Paris that Bradford Perkins published in 1983. The British were actually willing to adjust the Nippissing district's boundaries which would have given the USA what is now southern Ontario. The British were not necessarily scared to give land to the newly found United States in exchange for access to cheap raw materials.
https://www.jstor.org/stable/40105317?seq=6#page_scan_tab_contents
To summarize, the British were more concerned about making money than feeling proud about the expansion of their civilization. While Britons likely felt somewhat betrayed, they also saw many opportunities in the newly found Republic. In many ways, the establishment of the United States actually gave British investors reasons to cheer... that annoying Royal Proclamation legislation was nullified, they could still retain ties to their suppliers in the new republic, and they could still maintain a monopoly on most of the continent's fur trade.