When you were born, and without Googling or Wiki-ing....

[Supersonic]

Easy(er):

President of the USA - George HW Bush
Vice President of the USA - Dan Quayle
Monarch of the UK and other realms of the Commonwealth - Elizabeth II
Prime Minister of the UK - John Major
President or Premier of Russia or the Soviet Union - The drunk guy... err... YELTSIN.
President of France - Mitterand.
Pope of the Catholic Church - John Paul II

Somewhat more difficult:

Speaker of the US House of Representatives - Not sure.
Majority Leader of the US Senate - Not sure.
Prime Minister of Canada - Kim Campbell I think..
Prime Minister of Australia - Not sure.

Considerably trickier:

Governor of Massachusetts - No idea.
Governor of Illinois - No idea.
UK Chancellor of the Exchequer - Norman Lamont
UK Leader of the Opposition - John Smith
Most recent Best Picture Oscar winner - No idea.
Most recent Nobel Peace Prize laureate - No idea.

Very tough:

Senior U.S. Senator of Tennessee - No idea.
Junior U.S. Senator of Vermont - No idea.
Leader of the Scottish National Party - Alex Salmond?
Most recent Best Supporting Actor Oscar winner - No idea.
Most recent Nobel laureate in Physics - No idea.