LONDON — King Charles III is preparing to visit Australia and Samoa in October, an itinerary that will span 12 time zones and test the monarch’s stamina as he recovers from cancer treatment.

The trip, announced on Sunday by Buckingham Palace, marks a watershed moment for the 75-year-old king, who has been slowly returning to public duties after taking a break following his cancer diagnosis in early February. The decision to undertake such a long journey will be seen as a sign of…