The British Empire did not "cede" control of Colonies, between the late 19th and early 20th century the British gave the colonies the right to self govern and gave them the status of Dominion. This meant they could create their own parliaments and those parliaments could make rules affecting their dominion with a degree of autonomy. Representing the Crown was the Governor General, they held (and still hold) the executive functions of Government. Governor General where usually selected by the British Government (this was challenged in Australia in the early 1930's and lead to the first australian born governor-general Sir Isaac Isaacs) . The Canadian province of Labrador and Newfoundland (at its own request) was directly ruled from Britain from 1934 to 1949. Gradually when the British empire declined over the course of the 20th century. The dominions started developing their own national identities thus demanded greater autonomy. Thus the British slowly starting granting the dominions greater autonomy with the statute of Westminster in 1931 to the Australia, Canada and New Zealand acts of the 1980's. Australia, New Zealand and Canada are still very much legally tied to the British monarchy with the queen being still part of their executive branches.

OP is asking about why the British empire no longer controls its former colonies. The majority of the answers here seem to deal with the legal process, or rather how local rule was achieved.

So, what were the movements, the debates, the players, the parties, the rhetoric, etc that preceded the end of British rule?