Most nations have succession plans, but if 90% of the population has been destroyed along with much of the infrastructure (inferred by the use of WMD), then these nations have effectively ceased to exist.
The chaos in the rest of the world will have to be set aside for a moment (as the answer would become the size of an encyclopedia), but in the remnants of the various nations you will have pockets of survivors. The most likely pockets would be in military installations which were not affected, including deep command bunkers, nuclear and conventional submarines and deployed overseas forces. In this regard, the Americans are the in the best position since America has the most capability of any nation to project forces, and American units are deployed globally. Russia has a more limited ability to project forces and China, despite heroic efforts at modernization has a very limited ability.
Local commanders of these deployed forces have many options, since they have a supply of disciplined manpower and a lot of resources at their disposal. Depending on their personal inclinations, they could set themselves up as local warlords (directly conquering wherever they are), but the Americans, I suspect, will be more inclined to "buy" their way in using the broad range of technical skills combined with their military power to become "military contractors" for their hosts (mostly due to historical and cultural factors).
The Embassy personnel will be in a bit of a bind. Unlike their military counterparts, they have very little way to enforce their will, and many of the services they do offer are kind of pointless at this time (a visa to a nation which no longer exists? No thanks). Embassy personnel might choose to disband, or could hook up with their "national" military to gain protection but also provide the deployed units with intelligence and diplomatic support. Intelligence personnel "in country" will have to carefully establish linkages with their Embassy and military counterparts, or chose to throw in the towel and go native. National citizens who are stranded have a similar choice, and going to the Embassy to join their countrymen will broaden and deepen the pool of knowledge and resources available. Once again, the global reach of the Americans will put them in a much better position, since their people (tourists, businesspeople, NGO employees, etc.) are literally everywhere.
So what will they do? In analogous situations, exiles tend to form pockets in their host countries and live out their lives preserving their language, cultures and customs as best they can. "Little America's" throughout the globe backed by the remnants of US military power will most likely maintain a global community of American expats who will become global traders, "fixers" and middlemen, leveraging the huge commercial and cultural footprint that America has left across the globe. This may be something like the gradual decay of the Roman Empire, local Ducs change from being war leaders to administrators of the territories they can control, utilizing Roman roads and markets to maintain a tenuous Roman civilization. The Russians and Chinese will probably also have their own enclaves. Since there is a large Chinese diaspora, the Chinese will do better than the Russians, but the Diaspora will have distinctly different cultural behaviours from mainland Chinese, so any reconstituted China will be different than the current culture. Russian enclaves will not be so fortunate, being smaller and more isolated.
The end result may be a sort of ghostly pan global "American" presence, maintained by expat Americans with a cultural attachment to the symbols of America (the Constitution) and leveraging the American footprint, with a more tenuous Chinese presence built around Chinese culture. Russia has historically been built around a much more authoritarian culture and "personal" rule, fem the Tsars to Stalin to Putin, so I would expect Russian strongholds to be literally ruled with iron fists by Russian strongmen. Given the global chaos unleashed by the sudden demise of these nations, most "host" nations will have limited ability to eject the expats, and many will welcome them as extra resources to help ride out the storm.