Easy(er):
President of the USA - Bill Clinton
Vice President of the USA - Al Gore
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 - Yeltsin
President of France - François Mitterand
Pope of the Catholic Church - John Paul II
Somewhat more difficult:
Speaker of the US House of Representatives - Tip O'Neill?
Majority Leader of the US Senate - ?
Prime Minister of Canada - Jean Chretian
Prime Minister of Australia - Paul Keating
Considerably trickier:
Governor of Massachusetts - No idea
Governor of Illinois - No idea
UK Chancellor of the Exchequer - Ken Clarke
UK Leader of the Opposition - John Smith
Most recent Best Picture Oscar winner - Forrest Gump?
Most recent Nobel Peace Prize laureate -
Very tough:
Senior U.S. Senator of Tennessee - Fred Thompson?
Junior U.S. Senator of Vermont - Don't know
Leader of the Scottish National Party - Alex Salmond? The first time. But I doubt it.
Most recent Best Supporting Actor Oscar winner - Don't know
Most recent Nobel laureate in Physics - Don't know