Build a space elevator. The best place to build a space elevator is at the equator. So build it in Indonesia. I would tend to place it towards the western part. Palau Tanahmasa is probably around the right area, although the western coast of Sumatra is also possible. Funding might be provided based on the markets of Singapore (an island in that area). Independence might be maintained by playing off the United States of American and China. Neither is willing to let the other dominate this area, leaving the USO independent between them.
Since the space elevator will be the main way to launch things into space, this will give the country that manages it tremendous influence on space. The counterweight for the space elevator will operate under the USO's laws, and the USO can make it easy to launch space stations under their laws. This will give them outsized influence compared to their Earthbound population.
They can also act as the law enforcement of space. So when two space communities have a legal dispute, they appeal to the USO to adjudicate. The USO also could police travel in space.
Most things that require lots of electricity will move into space, where electricity comes from solar panels. This will give the USO outsized strength in the area of new energy production after fossil fuel burning is banned on Earth.
It's also worth noting that Australia's comparatively low population is restricted by its large central deserts. If it could desalinate cheaply, it could hydrate those areas, expanding its populable area.
Mining in space is likely to be done by automation in the asteroid belt. The USO could bring back that technology to Earth and use it to build saltwater greenhouses on the coast of Australia. In addition, a canal could be dredged through the heart of Australia so that saltwater greenhouses could be built all the way through.
Solar power could provide the electricity for this construction. Sun not shining? No work that day. But when the sun is shining, push on. This would be slow, but there's no real hurry. You're willing to wait a century.
Similarly, the islands of the area may be connected by saltwater greenhouse bridges. These will provide water and crops for communities that will build in what is now open water. These also will provide food for land-based communities built on what used to be cropland.
Like Australia, Antarctica is an island continent in the South Pacific. If its icepack melts due to global warming, it will be available for colonization. Adding it to the USO would make the coalition much larger. Again, the USA, Russia, and China may regard dominance by the USO as better than dominance by each other.
Another large piece of real estate to which the USO would have superior access is the Moon. To get to the Moon from the Earth, people will need to launch into space. And the USO controls the main space launch facility, able to launch cheaper than anyone else.
Indonesia is the fourth largest country in the world by population, after China, India, and the USA. Add in the Philippines, and it would jump past the USA. Traditional Oceania would add a more moderate amount, but that would still put it in the top four, even if the European Union was a single country.