Exiled royals go to the colonies
Wherein the home country in Europe is invaded, possibly by the French, and the royals have to go to the Colonies and continue from then on.
This AH cliché/overused trope is inspired by the exile of the Braganzas to Brazil between 1808 and 1822, after the Napoleonic invasion of Portugal in 1808. Predictably, these events had a massive impact on OTL 19th century Brazilian and Portuguese history, and often have a big or even bigger impacts in ATLs, if they occur.
A rather frequent variation on the general idea is that during an ATL WWII, the British royal family flees to Canada (as planned but not carried out in OTL, or that some anti-monarchical revolution occurs (republican-aristocratic, republic-democratic, or communist/syndicalist) and the surviving royal family has to flee to some royals-loyal part of the British Empire or British colony. Technically, the latter case did occur once in OTL history, when Oliver Cromwell banished the royals and made Britain into a quasi-republic/dictatorship, but the British royals fled to France and elsewhere in mainland Europe, rather than overseas, and the monarchy was eventually restored in the second half of the 1600s.
Other variations also exist. Rarer examples have even included the Russian imperial family being ousted from their homeland earlier than in OTL and seeking refuge in a still Russian-controlled Alaska, which they break away as a separatist Russian-speaking country, still loyal to the old Russian monarchy. There have also been French, Spanish and other monarchies in overseas exile.
Examples in AH.com and non-AH.com AH
- In EdT's Fight and Be Right, the British monarchy is overthrown by a syndicalist revolution in an alternate 1940 (becoming the world's first quasi-communist state). The British royal family flees to a royal-loyal South Africa, their new home-in-exile. Canada drifts away from its ties to the defunct United Kingdom and British Empire and becomes increasingly independent.
- In Tony Jones' Gurkani Alam (Mughal World) timeline, Novo Albion is a Russian colony in North America, a smaller constitutional monarchy that provides the now-exiled Russian imperial family refuge. Novo Albion considers itself the real Russia, rather than the theocratic Holy Russian Empire that overthrew the tsars after unsuccessful and ill-received reforms to the Russian Orthodox Church in the 1700s.
As you can see, though it is somewhat of a cliché, different authors have often sucessfully put an interesting spin on the basic idea.