I'm guessing that a third world war would probably be between NATO (North Atlantic Treaty Organisation) and SCO (Shanghai Coöperation Organisation) - for a matter of fact, America has probably already set off the chain of events leading up to this future war by creating such havoc and upheaval in the middle-east.
Currently, NATO consists of the Albania, Belgium, Britain, Bulgaria, Canada, Croatia, The Czech Republic, Denmark, Estonia, France, Germany, Greece, Hungary, Iceland, Italy, Latvia, Lithuania, Luxembourg, The Netherlands, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Turkey, and The USA
Currently, SCO consists of Armenia, The ASEAN, Azerbaijan, Belarus, China, Georgia, India, Iran, Kazakhstan, Kyrgyzstan, Moldova, Mongolia, Pakistan, Russia, Sri Lanka, Tajikistan, Turkmenistan, Ukraine, and Uzbekistan
What makes this interesting is that a war between the NATO and the SCO wouldn't purely be between the two sides. Most of the countries involved in the war would have other alliances and allegiances. One of the most notable being the European Union - if the European Union would be able to maintain peace among it's member states despite some being part of NATO and others being part of SCO, it could halt the war and turn a war between NATO and SCO into a second, much more intense and paranoid Cold War.
Of course, if the European Union isn't able to maintain peace among it's member states, it would be ripped apart, worldwide trade would be crippled, and there'd be a massive conflict between east and west europe with a potential death-toll of hundreds of millions. It would become the first world war all over again.
A war between NATO and SCO would put the British Commonwealth in an interesting position - many of the members of the British Commonwealth still maintain strong bonds with Britain and the other Commonwealth states. A perfect example of conflict of allegiances would be India, which are a member of the Commonwealth of Nations, but also a member of the SCO.
An interesting thought is that, with the commonwealth nations growing more and more isolated from the rest of the world, including britain, a "third" side may develop that could choose not to fight, or would have mixed allegiance if it did fight. Also, in my opinion, it would probably be in the best interests of the commonwealth nations for them to remain neutral, even with Britain's heavy involvement in the European Union and NATO, along with it's "Special Relationship" with the USA.
A full list of Commonwealth members can be found here - I choose not to list them here because of the large amount of countries in the commonwealth.
Lastly, Canada's position would be interesting. If Canada chose to fight with NATO, that would put them at odds with Russia, who are much more powerful and are practically right next to Canada. However, Canada chose to stay with the Commonwealth, having Canada as a neighbour could prove extremely risky for the USA, as the Commonwealth as an alliance would have a chance of switching sides between the NATO and the SCO at any point during such a war, not only giving the USA a next-door neighbour to battle with, but providing Russia with a direct route with which to attack the USA.
