Steven Den Beste has a long post mulling over what would happen to the Commonwealth if a no-longer-Great Britain were to lose its independence by being absorbed into Europe. Towards the end, he quotes an Australian reader named Russell:

If the Brits de-throned the Queen she would still be the Queen of Australia and separately Queen of some of our States as well (such as Queensland) - I am, and hope to always remain - a subject of the Crown. She would similarly remain the sovereign of large parts of the Commonwealth.

This sounds bizarre: how can a monarch rule (however symbolically) over a (former) colony if she does not rule the home country? However, similar things have happened before. In 1807 Dom Joâo, prince regent of Portugal, fled to Brazil to escape Napoleon. I don't know why it took so long, but he returned to Portugal in 1821, leaving his son Pedro in charge of Brazil. Taking a lead from Bolivar and other contemporary revolutionaries, Pedro declared independence from Portugal the next year and ruled as Emperor Pedro I. The Brazilian monarchy lasted until 1889.

Here's the Amazon blurb for a book I haven't read, Tropical Versailles : Empire, Monarchy, and the Portuguese Royal Court in Rio de Janeiro, 1808-1821, by Kirsten Schultz:

In 1807, to escape an invading Napoleonic army, the Portuguese Prince Regent and some 10,000 functionaries set sail for Brazil. Following the transfer of the court, Rio de Janeiro became a "tropical Versailles," a new world seat of European imperial power. In discourse and practice, residents and royal officials in Rio set out to transform a colonial capital into a royal court and to redefine the bases for the political legitimacy of an empire and monarchy centered in America. With the capital of the Portuguese empire in Brazil for the next thirteen years, an extraordinary inversion of political, economic, and cultural hierarchies that had governed colonial relations for centuries came to pass.

I don't know about the book, but the events it covers seem interesting enough.

There have been some other cases of empires continuing to function when the imperial power was no longer giving orders, e.g. the Dutch East Indies (now Indonesia) in the 20 months or so between Hitler's conquest of the Netherlands and the Japanese conquest of Southeast Asia. Local officials continued to govern, and Dutch armed forces resisted the Japanese invasion. The status of French colonies in Africa and elsewhere in the same time period was more ambiguous, since Vichy France itself was not exactly independent, but not entirely occupied by the Third Reich. I believe the Germans permitted the French to continue administering Senegal, Gabon, and all the rest. I'm sure someone knows how the administration was financed, though I do not. I assume Dutch forces in the East Indies were compelled to 'live off the land', with no subsidies from home, and no replacement troops expected. But I'm too lazy to look any of this up.