The answer is clear as to what New Vegas would become under the rule of Mr. House, Caesar, and even independence. NCR control seems to be the most mysterious, perhaps because we have no solid reference to them (we can't visit Shady Sands in 2081, for example). We only ever hear about their society second or third-hand, which doesn't paint a clear picture.

Under NCR it would be the 6th state of their union. Under Caesar it would become his "New Rome", his capital. Under House it would become his nexus for his rebuilding of humanity, his way and eventually colonizing other planets. Under the independent ending, everything stays the same more or less.

It would probably just become another state. If they don't fix up the poor area they may have some rowdy citizens, but nothing that time won't fix. It is probably the most mysterious outcome because it is the most boring. The NCR doesn't even want to fight the Legion so they will probably just go on ignoring them while maintaining strict border patrol.