The British Prime Minister needs to be an elected MP. She would need to be elected and then "promoted" for it to be an analogy.
They are not just "appointed", all the elected MPs get together and have a popularity contest as to who is going to represent them. That's all the PM does, they are not the head of state.
She is closer to the House of Lords, where people can suggest bills, and no one votes for them and even if they decide to be evil, it's virtually impossible for the people to remove them.
The House of Lords is also a terrible institution that is designed to remove power from those who vote.