Having grown up in a former British colony, I would say that the dismantling of the British Empire was probably unprecedented in its relative smoothness and lack of bloodshed. By the early 20th century, fractures were already appearing, and Canada, Australia, and New Zealand had already essentially become independent nations (excepting in issues regarding foreign affairs). As the tide of nationalism swept the world, all of the colonial powers began to recognize the difficulty in maintaining their empires.

No doubt the British wanted to maintain their empire, and India was the jewel of the largest empire in history. But not only did the Second World War weaken the administrative infrastructure, the British also recognized that if the Indians chose to pursue independence by force, there would be little that the British could do. Additionally, concessions were made to the Indians during the Second World War, including promises of independence in return for Indian assistance. They did try to negotiate for things like a joint governing of India between the native population and British administrators, but this was naturally rejected by leaders of the independence movement, including Gandhi and Jinnah.

Regarding the United States' involvement, the focus of America by this point was no longer the dissolution of the European Empires, but rather the containment of Communism. That the United States came to French aid during the war against Vietnam is proof of this.

As I mentioned, British colonies typically gained their independence from Britain without the need for war or violence: South Africa, India, Egypt, Malaysia, and Hong Kong are just a few examples. The British were brilliant in that they recognized that the old order was gone, and they had to adapt. To this end, they aided their former colonies in the transition to nationhood, establishing good relations and relatively stable governments, while allowing their economy and population to recover from the ravages of World War II. In this way, the legal and administrative structure of the former colonies also resembled those of the United Kingdom, facilitating diplomatic and administrative exchanges. With the establishment of the British Commonwealth, the British were able to benefit from this stability and friendliness through a lack of barriers to trade, providing the United Kingdom with many of the financial benefits that had existed prior to the empire's dissolution.