LONDON — King Charles III, who is 75 and battling cancer, will travel halfway around the world to Samoa this month to take his seat as the head of the Commonwealth and highlight the existential threat that climate change poses for Pacific island nations. He will also return to Australia, a country that played a key role in Charles’ adolescence — giving him the chance to be an almost normal teenager during the six months he spent at Timbertop school outside Melbourne in the 1960s.