*"In England they
just elected a new Prime Minister"

Far from it. The United Kingdom ( of which England is only one constituent part, along with Scotland, Northern Ireland and Wales) and its dependencies the Isle of Man and the Bailiwicks of Guernsey and Jersey have not had an election. We do not vote for an individual prime minister, but for Members of Parliament; the political party that has the most Members of Parliament at present - the Conservatives - have selected a new leader, namely Boris Johnson. We, the people of Britain, did not elect him.

It's a similar situation to what happens if a US President is for any reason unable to continue serving in the role, for example if he is impeached, becomes very ill, dies or is killed (hey, here's hoping) - a successor must be found to take on his role, but this is not done by a general public election. In fact, your method is more efficient than ours since the role is immediately given to the Vice President, or if the VP can't serve to the next figure down on a list of various government figures. Here, when a Prime Minister leaves the role, as when Theresa May resigned, the ruling party enters a long-winded series of contests in which any MP from that party puts their name forward and the members of the party have a series of votes to weed out those they least prefer until there is finally only one left.

The process was to elect a leader for the tories. They happen to be in power propped up by the dup. The process is noy over to install a pm. They still go tru ritual of visiting queen. Queen who is gods chosen leader. Then a parliamentary vote. Its hardly a good working model and certainly not a reresentative democracy.