I'm primarily interested in finding out how the Empire itself was structured in order to allow for quick and effective responses to issues that may effect a region, during an era where communication was relatively slow and the empire so widespread.
The centre actually had surprisingly little control over the colonies, with the notable exception of India. Disraeli complained that (I am paraphrasing) "we have all these colonies that we do not govern but we do pay for". He was referring to the UK paying for the defence of Canada and South Africa without actually deriving any taxes from them.
The situation was different in India, where the state had complete control via the Colonial office and the Viceroy of India was the British ruler of India in the name of the crown.
Good question! Is there any time period you are more interested in? The British Empire spanned for a couple centuries so zeroing in on a particular time period/technology level would be useful.