Of the places you mention I've only been to Palmerston and Nuie. Both were very interesting places to say the least. Notes from 2008
If you go to Palmerston the anchorage is pretty exposed and while they have moorings I would carefully inspect any mooring there and leave someone aboard who can manage the boat if it breaks. You may be the only boat there or one of a small group. Folks will come to greet you and invite you ashore. They have a unique history as I'm sure you have read. No resupply here. They will not care about customs/etc. A beautiful isolated atoll.
Nuie is a big limestone plateau sticking straight out of the ocean. Lots of boats stop here and there is a pretty good anchorage with several moorings. The only way to get ashore is to drive your dinghy to the landing area and then use the hand crane to lift it onto the plateau. Anything tied to shore will get bashed to pieces over time, there's no beach to speak of near the anchorage (or most anywhere else). You can bike around the island in a day and there are lots of neat places to explore. All marine activities are by boat because there's basically no shore, just cliffs. They do have an airport and get regular visitors and supplies by air so you can get most things here (even fly spare parts in from New Zealand when required). They are their own country so you will need to clear in/out though it is easy.