Final answer:
The British had complete control of the hill in India after the suppression of the Indian Mutiny in 1858, signalling the start of the British Raj. In America, the British took control of Breed's Hill after the Battle of Bunker Hill on June 17, 1775.
Step-by-step explanation:
The period when the British had complete control of the hill in India and the American colonies varies. After the Seven Years' War, the British East India Company expanded its control in India by defeating the ruler of Bengal in 1757 and capturing Pondicherry from the French in 1761, securing a significant portion of India. However, it was not until the suppression of the Indian Mutiny in 1858 that the British government directly took control over India, starting the period known as the British Raj.
In North America, during the early stages of the American Revolutionary War, the British sought to take control of strategic locations, such as Bunker Hill and Breed's Hill. The British achieved control over Breed's Hill on June 17, 1775, after a series of bloody assaults, infamously known as the Battle of Bunker Hill. The British suffered substantial losses but ultimately took the hill after the American forces ran out of ammunition.