Was Princess Royal already in the Caribbean?

Was Princess Royal already operating in the Caribbean during the opening of the war?

Not yet. She remained available to the Home Fleet in July-August and went to the Caribbean later amid concern about von Spee breaking through.

Princess Royal's later Caribbean duty falls outside the July-August force balance used in this counterfactual. She remained available to the Home Fleet during the opening crisis. Her westward deployment followed later, when British planners were concerned about von Spee breaking through.