Short answers, pulled from the story.
The Lord High Treasurer held a white staff, known as the white wand, which was the physical embodiment of authority. This staff was delivered to the appointee by the monarch and served as the sole method of appointment for the role. The staff was often made of ivory or wood and signified that the holder was the head of His Majesty's Treasury.
The English Treasury emerged around 1126 during the reign of Henry I. Before this separation, financial responsibilities were merely a section of the Royal Household. The office took a more defined shape in 1216 when a Treasurer was appointed to take control of the Treasury in Winchester.
William Cecil, 1st Baron Burghley, served as Lord High Treasurer from 1572 to 1598. He was the right hand of Queen Elizabeth I and effectively ran the government. Cecil controlled the flow of information to the Queen and managed the kingdom's finances with an iron grip.
The office of Lord High Treasurer has been vacant since the resignation of Charles Talbot, 1st Duke of Shrewsbury in 1714. When the office was left vacant, the Consolidated Fund Act of 1816 provided that the monarch could appoint Commissioners to execute the duties. These Commissioners became the mechanism through which the functions of the Treasurer were carried out.
The Prime Minister of the United Kingdom is appointed as the First Lord of the Treasury. This position carries the ceremonial title of Lord High Treasurer but is exercised through the commission system. The Chancellor of the Exchequer serves as the Second Lord of the Treasury while the Prime Minister holds the overarching authority.