Post-Kaiju Earth is going to be a real basket case for a bit I imagine. The economy is in a shithole, people are working for food, the last good job market (Wall of Life) just dried up, there are riots in the streets and most pacific coastal nations are going to feel abandoned by the elite and powerful. Couple that with angry Kaiju cultists, first contact with another alien species and weak political leaders and things are all adding up to 'interesting times'.
This of course is not a recipe for stability and happy times. I'd like to keep politics out since that's likely to initiate a flame war, but speaking in broad terms, Post-Kaiju Earth shapes up as follows:
Australia, New Zealand and the pacific islands merge under a new entity called Oceania.
It's likely that Australia and New Zealand took in pacific island nations that were succumbing to Kaiju assault. Those that hung on likely have their economies in the toilet, and merging with a larger local power is preferable to becoming a failed state. The fact that Striker Eureka was one of the most powerful and successful Jaegers and in no small part likely saved a majority of the Pacific by its mere
existence likely also bought a lot of good will, and a generation of refugees who grew up on AUS/NZ soil will have a fair amount of political clout. Indonesia remains independent and serves as a buffer between Oceania and China.
China continues to rise as a global power.
The Kaiju war and Crimson Typhoon only helped things along. China doesn't have much pacific coastline even with Hong Kong in the picture. The fact that they likely took very little damage from the Kaiju War means that their economic and agricultural might is largely unchanged compared to other nations. Their continued colonisation of Africa is also likely paying them dividends. On the downside, global instability isn't exactly a boon to their economy, and the downturn globally has affected their manufacturing industry. They also have other headaches locally.
America either fractures completely or just barely manages to hold together.
This really depends on the politics and the president of the time. If the President is a firm supporter of the Jaeger program, is charismatic, and was largely overruled by a hostile congress, then its possible that they might be able to pull America over the line and through the tough years over the bodies of their political enemies as they get recalled and elected out of office for their monumental failures. If the President supported the program but isn't charismatic enough, the riots continue until they're ousted and replaced, likely by a massive populist who will want to bring the Jaeger program back into the fold and use it as an emblem in their political machinations to hold the nation together while securing their political future.
If however the president was against the Jaeger program with Congresses backing, the riots that were shown in the film are likely to intensify. The coastal pacific states are likely to be incredibly hostile to the federal government that abandoned them, likely calling for their resignations and a new round of elections. If that doesn't pan out, California is likely to secede, followed by the remainder of the Pacific states and their neighbors. Once the avalanche starts, the south likely joins them, leaving three distinct regions of East, West and South, with the odd few states opting to join Canada. Whether this descends into a three way civil war or merely a very tense and divided America is up to the imagination. Long term though, fences are mended and unity established in principle under an EU-like federation, but the regions remain distinct for a while.
The likely scenario is somewhere in between though. Given that the American representative seemed eager to kill the Jaeger Program during his brief appearance, I'm leaning towards a lame duck president being more likely than a charismatic one that was overruled by congress, which lends credence to the latter scenario with elements of the former. America is likely to draw inward as it sees to its own problems, which means a return to isolationism as they try to fix the fractures in their own backyard.
Japan also returns to isolationism.
Surrounded by neighbours who wouldn't take them in even if it
was the end of the world, Japan is in an odd spot. They have a declining population, a tough as nails immigration policy, few natural resources, and were likely hit hard by the Kaiju. On the other hand, they likely have the best experts for robotics and making Jaegers. America's decline in the face of internal fracturing means that the defense pact that keeps them nominally safe from attack is yanked out from under them. This leads to Japan likely remilitarising, but without the population of young people to turn to, they opt for a robotic and drone army to keep their borders safe. China and South Korea are not pleased with this development, but with the withdrawal of America there's not really much that can be done short of keeping the Japanese honest to their constitution that their military remains largely defensive in nature.
While remaining economically linked to the rest of the world, Japan likely becomes politically isolationist as time wears on. Whatever winds up representing Earth in the near or far future, Japan has little to do with it politically if anything at all, and probably less than that once the stipulations against robotics and artificial intelligence becomes apparent. They still back the Jaeger program as their expertise remains ever increasingly firmly entrenched in robotics, and their indispensable nature on that front likely earns them a blind eye from the other Earth nations who might otherwise be persuaded to pressure them into falling in line. How intelligent their robots become as time wears on and how useful they are when they meet the Geth is decent question. This is the nation that gave us Chobits, Ghost in the Shell and Astroboy after all.
Europe becomes a prominent power again.
Untouched by the Kaiju war and the center for the development of Jaegers, Europe sees a resurgence of their power in the century to come and becomes a major player in presenting a unified front for the world.
Russia is Russia.
What more is there to say? They didn't take much damage from the Kaiju War. A few eastern European states get gobbled up but that's about all that changes. Possibly a return to Tzarist rule in the long term. Probably butt heads with Europe and China, but that's also nothing new. Massive shit-eating grin as America goes through its troubles. Huge point of national pride that is Cherno Alpha.
Argentina likely tries something with the Falklands while everyone is distracted. Gets their asses roundly kicked in. Again.
Iran likely becomes a major power in the middle east with the withdrawal of a majority of American interests in the region. Israel becomes the likely, if hugely disliked, counterweight to Iranian power, with Turkey, Egypt, the UAE and other nations being players as well. The region remains a hotbed of instability forever
due to reasons and likely doesn't get too involved in interstellar affairs with local problems keeping their attention firmly at home.
India will also prove to be a rising power politically speaking, with a massive chunk of humanity under their belt on one hand and competition with China on the other.
So, rounding that out, politically speaking:
Major Players:
Europe
Russia
India (rising power, major stake in future space colonisation plans)
China
America (declining / civil war near term, returning to parity long term)
Minor Players:
Oceania
South Korea
Taiwan (if not absorbed by China)
Japan
South Africa (resources, rising power)
In terms of the Jaeger program though:
Major Players:
Europe (R&D)
Japan (Robotics Expertise)
Oceania (Striker Eureka. 'Nuff said)
Russia (Cherno Alpha. 'Nuff said)
Minor Players:
South Korea
China (Economic backing)
Taiwan (if not absorbed by China)
South Africa (resources, rising power)
America (declining / civil war near term, returning to parity long term)
India
This is all very broad strokes and invokes a lot of Hari Seldon. Individuals can and will make a difference, and it's very probable that I don't have an exactly masterful grasp on the nuances of the politics of the various nations presented here, but this seems to me to be the likely outcome of a post-kaiju Earth.
All that being said, the likely representative for Earth in the near term already exists in the United Nations. That's pretty much the go-to option until something bigger and better can replace it, and that's unlikely to happen unless a majority of unity is demanded, such as say, Batarians deciding the humans will make great and unique slaves. In the long term, if Earth and her colonies do manage to largely unite under a single banner, the name is likely to be something that unites not only Earth nations but also her Colonies as well, which in turn will be determined by who has the power and who has to be brokered with and what style of interstellar government they want to be a part of.