As the Western Hemisphere comprises countries to the west of the Prime Meridian, I think more than just the "new world" is involved e.g. Ireland, Iceland, Portugal .....
+1. I live in the Western Hemisphere (just about - I can see the Meridian Line marker from my house) in the UK. I would not say that I live in the New World, which was mainly "discovered" from a European perspective by the UK, Portugal and Spain, all of which are substantially or exclusively in the Western Hemisphere
I live in the Western Hemisphere - Ireland. Since when did an entire hemisphere relate only to one particular landmass? Did the Western Hemisphere not exist before 1492?
For the neighbour question I put in "What in the world.." It is what I immediately thought of and seriously thought you were after. I think it fits well enough to be an acceptable answer.2
It’s also the name of the underworld itself. Like Erebus, the underworld that also shares a name with the god of darkness. My kids are super into the Percy Jackson series and it spiraled from there.
Australia is seen as aligned with the first world and China was considered 2nd world.
They did not need to be "in" the NATO or Warsaw pact.
You probably are thinking of "developing countries"
https://en.wikipedia.org/wiki/Third_World