When Did Britain's Kings Lose Their Power

The key event was the Act of Settlement in 1689 following the 'Glorious Revolution' of 1688.. James II had been bloodlessly booted out and his (protestant) daughter Mary made joint monarch with her husband, William of Orange.
The Act of Settlement included a Bill of Rights that required that no Roman Catholic could become monarch; it declared illegal the monarch's power to suspend the operation of laws passed by Parliament, and made Parliamentary freedom of speech absolute. The 'Toleration Act' also gave legal freedom to Nonconformists (Trinitarians only) to worship according to conscience. These freedoms were limited. I can go further into that if anyone is interested.

In 1714, Mary's sister, Queen Anne, died without issue and her nearest Protestant heir was a German, George, elector of Hanover who became George I. He spoke no English, and so political matters passed further into the control of Parliament. The First Lord of the Treasury became known as Prime Minister, the first one being Sir Robert Walpole who ran affairs on behalf of the king..