I might do several of these based on the UN/modern-day, but let's say a coalition of the following countries begin a worldwide police action... the Arab League, Brazil, Chile, Peru, the Commonwealth (minus the UK, Australia, Canada and New Zealand), the Association of Caribbean States (minus Mexico, Costa Rica and Belize), ASEAN, the CIS (minus Russia, Moldova and Belarus), China, North Korea, Afghanistan and Iran... to compete with the United States and NATO allies
The world descends in World War III, with the West declaring war on the UN and global outbreak of hostilities beginning in the failed state of Haiti (because I know for sure they could defeat and conquer Haiti) and spreading across the planet. Assume that the UN soldiers and police are superseded by national armed forces and police, with the full weight of each country's respective economy marshalled against whichever countries they target. Also assume the UN countries maintain relative stability throughout the duration of these conflicts in non-coalition countries. In which region are they bound to start having problems? Which members of the coalition will pull their weight and which won't?
Hmmmm if Russia is feeling threatened, it might be persuaded to angle for the Western side, however the OP states that the United Nations is declaring War on the West, so it might be more likely that Russia leads a third way bloc.
So you'd have the West, with what appears to be the entirety of NATO and the European Union, plus likely ANZUS, South Korea, Ukraine, Moldova and Japan. Maybe Switzerland as well, but a minor issue. Taiwan might also be involved since they could be roped into 'The West.' Maybe Georgia and Armenia as well, but they might also want neutrality to avoid being on the front line. The only other issue might be Ethiopia since they could be open to pressing claims for the Nile River and/or Somali but it seems doubtful considering their geographical location as well.
Russia would likely be leading with Belarus and maybe some of the West African states, Central African Republic and Zimbabwe and going their third way. Possible Serbia as well but they would probably like to be double neutral.
Countries like Nigeria, Mexico, Argentina and the like would likely just want to stay the hell out of affairs in general, no alliances in general. But Mexico would probably be leaning towards the United States in sympathy since they are on a borderland.
Immediate fronts would be the Caribbean, the Southwest Pacific, Turkey and the Northwest Pacific. I think the Caribbean Front would be hard pressed to hold back the United States and Canada and a lot of those islands would be in dire straights soon enough when the blockades and airstrikes start taking their toll. I don't think the Latin American air and naval forces will be able to intercede well enough. Cuba would become an island in an enemy lake pretty quick I feel.
North Africa is powerful but there is a vulnerability in the reliably of most of the transportation being along the coast along with industry and other economic and population centers. I don't think they'll be able to engage in any cross straight or sea operations and be punished quite a bit by the European forces there as well.
Turkey would get hammered pretty tough but the terrain is mountainous and defensive. They are up against some formidable peer opponents so it'll be tough for them but with European backing and Ukraine as well, I think they could hold and stabilize.
Australia looks to be in a bad position but it's military is large enough I feel to keep their shores safe for now. The real Battle I feel will be for the Northwest Pacific. Taiwan and Japan and South Korea and everything in between.
As far as pulling weight, I feel that the Central American and Caribbean states will actually collapse far more quickly morale wise. I don't think they'll be wanting to fight this battle. None of them are really expansionist and the odds in the local theater are really long against them. Plus collaboration will be strong, as well the economic opportunity that peace will bring. Latin America is similar but would require more pressure to fracture I feel. Especially countries with significant revolutionary and political divides like Colombia, Venezuela and Peru etc or issues with other countries. Like I don't see Chile wanting to sacrifice too much for the UN, weakening them if Argentina and/or Bolivia or something decides to pursue opportunistic revanchism in this zany alternate world.
In the Pacific, I think the Solomon Islands and Papua New Guinea would actually fold. They don't have the depth and would need lots of support to sustain pressure from ANZUS and allied forces. As the Western Hemisphere stabilizes for the West, I think that Indonesia and Philippines will then feel the pressure of supporting the United Nations with the former thinking of wanting to pursue surrender and neutrality, and the latter switching sides to capitalize. A lot of the SEA countries would see themselves as actual battlefields for great powers like China so that China can benefit from gobbling Japan, Taiwan, Australia and whatnot, not themselves. The prior rivalry and issues with China would strain their relationship with ASEAN allies as they realize they are increasingly the sacrificial buffer zone to protect China and I think that's a fracture point in general. They'll fold like the Axis powers allies in the Balkans did IMHO.
Taiwan would have a tough time, then South Korea... and then Japan. It'd be a devastating conflict for all of them but the fact you have three powerful countries means that the extremely formidable power of a Sino-Indian Alliance would be split among them... and also require attention on the United States and ANZUS as well. Even if they can capture some of the Islands and invade Taiwan, I don't think they'll be able to secure it when their ASEAN allies decide not warring with the West is in their better interests and they have more to gain from a weakened China instead to press their future claims against.