There were three Australian Prime Ministers during World War 1:-
(1). Joseph Cook (Liberal party) from 24/6/1913 to 5/9/1914.
(2). Andrew Fisher (Commonwealth Labor party) from 5/9/1914 to 27/10/1915.
(3). William Morris ("Billy") Hughes (under various parties) from 27/10/1915 to 9/2/1923.
(He was Prime Minister for the Australian Labor Party until November 1916, then for the minority National Labor Ministry until Febuary 1917, and from Febuary 1917for the National coalition with other conservatives.)
John Curtin - to participate in the defence of Autralia after the Japanese entered the war in 1941, rather than leaving them fighting in a European war.
Peter Fraser .
In Great Britain, it was David Lloyd-George.
No future Australian PM has been shot down.
Winston Churchill was PM during WW2 : 1940 through 1945.
From 1911-1920 the Prime Minister of Canada was Sir Robert Borden.
Peter Fraser was the pm of New Zealand during WW2 however, New Zealand did not take part in the war.
In the UK, Neville Chamberlain
H.H. Asquith
Joseph Lyons died in office before WW2 had started. Earle Page was Caretaker Prime Minister for 19 days and Robert Menzies was subsequently elected Prime Minister. It was Robert Menzies who announced that Australia was at war.
Clement Attlee
King George VI was king throughout World War II.