Brazil and the Caliphate need to find a way out of the system. Both because of the disaster setting and mineral depletion. Unfortunately this is difficult because of all the territory claiming going on. Mars and the Centauri Federation are in less dire need, but have to know they can't stay in one system forever. Of those four, only Brazil has any military to speak of, and isn't going to be on speaking terms with the Centauri or Mars. This means a diplomatic approach is probably going to be necessary for them to find new places to expand. Additionally, the Colonial Wars proved that alliances are necessary; Brazil was the aggressor and the Caliphate were the defenders, but both lost because multiple nations moved against them. Any future wars are going to require a network of alliances to make sure that doesn't happen again in order to have any real hope of success.
The Commonwealth is in a good situation; they have their own territory, a powerful combined military, and recent tech discoveries that promise to help them secure both. If they ally with anyone, the Caliphate would be well served to pursue a relationship with them, since they have what they need (new locations to expand to, and sufficient military strength to keep away vultures while they rebuild their shipyards and military). It also works out well with the Commonwealth coming off as something of the "good guys" since the Caliphate are definitely the kicked puppies at the moment. Since the Caliphate got to recover its ground forces and troop transport, they could even work out a deal where they handle the Brazilian colonies that the Commonwealth is annexing, which would lower the Commonwealth's Badboy score if this were a Paradox game :p
I agree that Brazil and Manchuria are natural allies. Brazil needs territory, and Manchuria has territory but needs a military capable of giving Russia pause. Given how paranoid Russia tends to be, this could set up a cold war scenario, with both forces disliking the other but neither confident in their ability to wage a decisive war.
I expect Mars and the Centauri to grow closer. Mars didn't accomplish much, but they did try to stick with the Centauri even when things were bad, which will probably bring them closer together. The obvious ally for them would be Japan; Japan intervened in the conflict, and controls the rest of the Centauri arm. However, Japan's intervention was nakedly opportunistic, and I doubt they'd let the Centauri colonize their space after demanding unilateral control of it as part of their price. So I expect Centauri/Mars will have to look elsewhere.
France has lots of space, but relatively few good colony sites, so unless they're up for sharing the same worlds they might be reluctant to allow colonization in their space. Germany and the US are probably more interested in building their own alliance. The Commonwealth is already pretty large and has very little to gain from a Centauri/Mars alliance, since they'd have to split their military if they wanted to secure the Centauri system.
Russia would, oddly enough, be a great ally; they have lots of territory but limited ability to exploit it. On the other hand, Russia tends to be paranoid and xenophobic in these games; they're probably not going to be willing to share. Since Russia has no real reason to mess with anyone and no one but Manchuria has a reason to mess with Russia, I predict they may grow increasingly isolationist. Also until the Centauri can rebuild their military they have little to offer Russia as far as defense against Manchuria.
That leaves Argentina, but I'm not sure they'd move fast on this one. Argentina's space is far away from Earth and Centauri, and the resources of the Centauri system leave them in good shape for now. I'd say the Mars/Centauri alliance might start trying to work out some initial deals with Argentina, maybe a trade treaty of sorts. Meanwhile the Centauri will be rebuilding their military while trying to keep their options open.
