Leaving aside pointless digs at other nationalities (no offence meant, adam, but the snide comments I see so often here about the French just grate after a while)…
It’s a really tough question to answer, purely because it’s so obviously difficult to imagine the alliances involved, and public pressure to avoid futile, bloody wars would play an immense role in the US and in Western Europe.
Still, forced to make some WAGs, I’d say the US would lose in the end. Here’s what I could see happening:
Defending US borders
No immediate threat of a land invasion, but Central and South American forces massing on the southern border draw large numbers of Continental US troops south in readiness. Canadian forces attack and, despite vague technological parity, are quickly overwhelmed by US airpower and numbers. Of course, US forces have to patrol the northern border and some Canadian raider units penetrate into the northern states, causing terror among the local population and preventing the US from redeploying their northern forces elsewhere.
US forces in Europe are destroyed/captured after bloody battles. Sorry, but the quality and technology would be no match in the medium term when you consider the US would have no re-supply or maintenance facilities. British, French and German armour takes heavy casualties but soon grinds down the US troops in the West. Add in troops from the former WP countries and the US forces in Europe are stretched to breaking point fighting off attacks from all sides. Only one result there. (I’m assuming US airpower would go the same way, if not faster as British, German and Russian commando forces lay waste to airfields in Germany and the UK).
In terms of forces, then, the US has lost its European forces entirely. Western and Eastern European nations will have bloody noses but won’t be shattered by any stretch of the imagination.
China and Russia, with added manpower from India, mass a naval force to attack the US west coast, spearheaded by every submarine they can muster. Amphibious operations don’t start; none of the nations there have a great Pacific amphibious capability, and all know the lethality of the US Navy. US Navy forces in the Indian and Pacific Oceans score massive victories over all-comers, but are slowly and inevitably ground down by wave after wave of submarine and air attacks. I know a MiG-23 is no match for an Aegis-equipped task force, but throw enough of them and enough SSMs and you’ll be out of Standard missiles eventually. US forces in the Indian Ocean survive longer by playing a little cat and mouse, but are rendered ineffective in defending the US in their desire to stay alive.
Off the US west coast, US forces stop everything. The allied attackers suffer grievous losses at sea and in the air, but not without inflicting some damage. If the US loses one, maybe two aircraft carriers, the situation changes dramatically. Terror attacks on US cities and dockyards on the west coast hinder the US Navy’s ability to repair and regroup. Eventually, as Chinese/Russian subs and surface vessels sacrifice themselves to tie up the US Navy in the Pacific, land forces are landed in Alaska.
Alaska falls quickly: Canadian raiders draw troops east and the threat from the west occupies the others. The advance is slow (I’m sure it’s not easy to transport and land troops that far north) but inevitable. Russian airborne divisions leaprog ahead as Chinese foot infantry divisions negotiate the tricky terrain. This draws US forces northwards to engage through Canada, further stretching the regular Army and NG units and exposing them to guerilla attacks as they move through Canada.
At this point, Central and South American land forces attack en masse. US air and land forces win heavily to begin with, but every attack they repel takes strength away from the air forces supporting the Navy in the west. The US holds the southern border, but only by re-jigging its forces constantly.
The Russian and Chinese naval forces are (by this point) in tatters, and what’s left withdraw north to protect the transports landing in Alaska. Chinese divisions are shipped across en masse (foot infantry, no armour) and Russian armour slowly arrives later. Bloody stalemate ensues, as neither side has the ability to dislodge the other (allied numbers vs US technology).
In the east, European forces launch naval attacks to sap the US Navy’s Atlantic and Med forces. Bloody stalemate, again. Constant air and SSM attacks grind down carrier air groups and force the US surface vessels closer to home for resupply and replacements. US submarine forces stop anything getting through, but in doing so are prevented from reinforcing the Pacific (and anyway, South American forces have mined the Panama canal).
After that it becomes a war of attrition, which the US cannot win in the long term. Technological superiority disappears – and a Seawolf-class SSN, Nimitz-class carrier, F-117 or B-2 can’t be replaced quickly. As US NG forces and older equipment fill the gaps in the line, allied forces grow more confident. European attempts to finish off the US Navy in the Atlantic redouble. Heavily protected European convoys land troops in Canada for a slow trek down to the US border, and another assault. By this point the US is facing a land war on three fronts, a sea war on two coasts and diminishing technological superiority.
So there you have it.