They would settle in NZ & AU, with a marine army defending against the rest of the world, which would be busy anyway fighting for survival against one another and climate. Use latest tech to keeps these lands habitable and self-sustaining. Maybe even expand to a space station in 50 years.
If we look for examples of similar bunkers, there are a couple military sites but nothing (AFAIK) that's been inhabited long term. Their concerns about keeping the guards on their side seems valid, that could turn out to be intractable. If the whole world does crumble, they'll have to be able to build everything from raw materials, including sourcing all of those raw materials. And so on and so forth.
Why would the existing populations of AU & NZ consent to becoming the subjects of some billionaire aristocracy? We're talking about millions of people formed into actual nation states with sizable and competent militaries. Not even the combined private security forces of 100 billionaires and 1,000 millionaires could take control and dominate something that large. Elon Musk isn't going to be able to wow them into submission with minisubs and beads.
I agree that setting up their own little city out in AU or NZ might be a better idea than the bunker. Maybe they can even take over a large section of the land mass.
In my opinion, it'll still be tough to fabricate all of the things that modern society depends on today. Maybe they can do it if they really put their minds to it, but I would be surprised.