I'm pretty sure on this one, I think that every country with nukes launched nukes and every other country. First it was just USA and China, either USA's allies attacked China's allies, or China's Allies attacked USA's allies. It's safe to assume every country was nuked at least once.
The US's allies generaly had their own problems, and probably weren't too friendly with the US at the time.
The US had the last source of oil, and wouldn't share it - THis caused the UN to break down.
Europe (and therefore the "NATO" allies (Assuming "NATO" existed... But either way, the US's WW2 allies)) had dissovlved into squabbling city-states.
Canada was busy resisting the US invasion/annexation. Other countries, even allies, tend not to like annexations.
Australia and NZ traditionaly do not have nukes (WW2 caused both Australia and NZ to fall out of the UK's defensive sphere of influence and into the US's, so its reasonable to assume they're with the US), but you're not looking at too many people troopwise to help out there, assuming they had the oil to get some troops anywhere.
I think its fairly safe to assume the US was more or less on its own.