The US are by far not the only ones with satellites in orbit. Making it a UN body would make sense, just like the International Telecommunications Union coordinates telephone service and the International Postal Union coordinates international mail, and both are now UN bodies (despite predating the UN).

I have a feeling that the current US administration would not back such an idea, so this will end up back with the DOD, maybe the Space Force. Despite the DOD saying quite clearly they would prefer NOAA to do it