Iran and Israel, two of the most ruthless nuclear capable countries in the world, will start a shit storm -- they'll invade the countries around them unti they border each other and then form a miniature cold war. These two countries become the centres of power in the middle east, controlling essentially all of the remaining oil in the world.
Most would say "well, the yanks back the Israelis so they would win with superior technology", but lets say an up-and-coming economy like China needs more and more oil but can't meet demands. They can't make an agreement with Russia and the USA has their claws in Israel, so they turn to Iran.
Pretty soon things heat up massively as the nut-jobs who run them begin stockpiling and making outrageous claims on oil in areas that their opposite number has already occupied. The UN will try to reach some sort of peaceful solution, but with the US and China gone it loses much of it's power.
The UK will finally be forced to choose whether to support the US or the EU after toe dipping n both for the past five decades. India will most likely back America and Britain, odd as it sounds, because of their awkward relations with rival power China and their love of western culture that dates back a century.
The EU will actually be the most potentially destructive superpower on the planet, with the combined nuclear threat of three of the world's biggest economies plus many more minor ones as well as some of the largest standing armies going under it's wing, however because of the mistrust that even still the EU treats it's allies with they won't make a move.
Some shit will start in the balkans between the EU and the Russians over the pipelines that run through them and the small oil reserves there, heightening tensions further.
All over the world countries will become more and more tenacious in the search for oil, and the Russians will once again become a very closed state, re-building their armed forces and allying with China over the west, backing Pakistan as a threat to India and engulfing the surrounding countries such as Kazakhstan and some parts of the odl eastern bloc, who will welcome the kremlin back. Russia will quite possibly have another communist revolution, though this time their regime will be much less harsh and have more trade with other countries, though this will still reduce massively.
Japan and S. Korea will back the Americans whilst N. Korea will strongly back China
Israel and Iran will nuke the hell out of each other. Something else will set off N. Korea, who will attempt a pre-emptive strike against S. Korea and Japan. Russia will invade the balkans and fight with EU forces there, but neither will fire nukes yet.
In South America they'll have their own little shit storm going on between Brazil and Argentina, who have grown into rival economic powers.
China and America will try one last time not to laucnh their nukes, but some sort of lapse, possibly an accidentally sunk ship, will completely destroy communications, and suddenly it will be a race to see who can fire their nukes first: The US and China will destroy each other utterly, with Canada and the countries of south east asia getting caught in the crossfire.
The Russians will finally fire a nuke into India, on their new found border with which tensions have been steadily mounting for years. India will launch it's load at Russia, however their fairly weak nuclear programme will not be able to stop the kremlin, and then beg for help from the EU, who will oblige, bombing several major east russian cities.
So kind of in short:
China and the USA have blown each other to shit
Russia and India have blown each other to shit
the EU's diplomacy lies in ruins and tensions within it are high. Will possibly destroy itself in the coming months.
S. Korea, Japan, N. Korea, Israel and Iran are no more.
Most of the middle east has been devastated by fallout but has also been tearing itself apart before this.
South America will get into it's own little world war.
Africa has had old school wars and invasions with some of the worst atrocities occuring, but their lack of a nuclear deterrant has turn the world's eyes away from them.
The Balkans are dust, so is much of eastern Europe such as Belarus and Poland.
The EU three (UK, France+Germany) just survive, though their farms are utterly ruined by fallout from Russia. Southern Europe (southern parts of Italy and Spain) will escaoe unscathed.
Australia will have become a missile base for the US, like Britain did in the 70's, and as such will have been blown to hell by the Chinese.
Your best bet is southern central america, most likely. That or space.