do you have to be staying at animal kingdom to gain access to the Nomad Lounge? or is it open to public

You must have a park pass. Then you will be required to walk up to the host, if there is a wait they will require you to be added the list. There is no online reservation system.

Must have a valid park admission; ticket and park reservation for that day. Than when your in the park you can join the walk-up waitlist, which at 1130am on monday was only a 7 minute wait, or reserve your spot on line on the my disney experience app, but you need to be inside the animal kingdom park to get into nomad lounge.

And a reservation for the park plus a ticket.

Open to the public, but since it is inside Disney's Animal Kingdom, you'll only need a park pass.

Public

You have to buy the Animal Kingdom pass or you Could buy any Other but you have to add the hopper option to be able to get there .