Vernon Dursley once doubted that Dudley even knew who the Prime Minister was, since the boy took no interest in current events, but neglected to mention the name himself (OP1).
Vernon would be horrified (judging from his reaction to the existence of the Ministry of Magic in (OP2) to know that each Prime Minister is made aware of the existence of the wizarding world upon his or her election, and subsequently has contact with it in emergencies. In recent years, the Muggle Prime Minister has been notified of Wizarding emergencies four times (HBP1):
- Sirius Black’s escape from Azkaban (PA3)
- torment of Muggles at Quidditch World Cup (GF9); Fudge also used that visit to inform the Prime Minister that due to the Triwizard Tournament, three foreign dragons and a sphinx were being imported into the country
- the mass breakout from Azkaban (OP25)
- events related to the breakout of the 2nd Wizarding war (destruction of the Brockdale Bridge, murders of Emmeline Vance and Amelia Bones, rampage by giants, failed curse on junior minister Herbert Chorley) (HBP1).