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How about the ouster of Reza Shah Pahlavi in favour of his son Mohammed Reza Shah Pahlavi in 1941, by the British and Russians.
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British gave support to the revolutions in Spanish colonies in America. Here is the Spanish article of that strategy in the Wikipedia, because after the British military defeat trying to directly take by force those territories they decided to simply support revolutions.
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Brief review of wikipedia leads me to conclude that it would be more accurate to say that the British government could furl the winds of revolution. The British government repeatedly (e.g. Oman, or @T.E.D's example of Egypt) provided support to a ruler who would otherwise have been at risk of overthrow. That said, below are a few examples of British support for rebels or counter-government forces that ultimately benefited the Empire.
The British, based in Calcutta, supported rebels from Manipur, Assam and Arakan fleeing into British territory. Wikipedia:First Anglo-Burmese War
Meanwhile, in 1793, France declared war on Great Britain. The white planters in Saint Domingue made agreements with Great Britain to declare British sovereignty over the islands. Wikipedia: The Haitian Revolution
Siraj-ud-daulah had become the Nawab of Bengal the year before, and he ordered the English to stop the extension of their fortification. Robert Clive bribed Mir Jafar, the commander in chief of the nawab's army, and also promised him to make him Nawab of Bengal. He defeated the Nawab at Plassey in 1757 and captured Calcutta.3 Wikipedia: Battle of Plassey
One might argue that Great Britain didn't "unfurl" the flag of revolution, but merely allied with an ongoing revolution. I submit that is a distinction without a difference.
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I don't think this was done 100% on purpose, but that's roughly how Egypt ended up part of the British Empire.
The ruler of Egypt in the mid-1800's was not very good with finances, to say the least. He ended up having to sell the Suez Canal to the British, and things continued to deteriorate to the point where the entire country was essentially put into receivership.
This inevitably sparked an anti-European uprising, which in turn threatened the (largely British) financial interests to whom the country was in debt. That in turn got the British Navy and army sent to restore order.
Again, the UK didn't start buying Egyptian assets and debt with the clever master plan of eventually taking over the country. However, the effect was certainly the same.