Hi! If you can get your prereq's done while you are overseas, I would do that, because once you PCS to the states you will have to be on station for one year before you can apply. If you apply from overseas, they can curtail your DEROS for you (that is what they did with me); makes it easier than having to wait another year.
I love Psych as well, so mental health doesn't sound bad. But I am leaning more towards Neonatal or Surgical.
You have to apply to colleges before the NECP deadline because you have to have documentation from the schools that you have been accepted, either officially or conditionally, in your package. You also have to have your Form 48 filled out by the school that demonstrates that you can finish the program in two years. Every college has a different deadline, though, some are very early (November of the year before) and some are very late (June).
I bought the ATI Testing TEAS V Study Manual as well as both versions of the practice assessments from the ATI website. I would not recommend not studying for the TEAS, the questions can be tricky and the study guides get you ready for the types of questions that they will ask on the test. Don't bother trying to memorize the practice tests or the book though, because the questions aren't the same. They are similar, but if you simply memorize the answers but don't understand how you got them, you probably won't do very well. I took Version 5, so that is all I can speak for. I hear that Version 4 is easier. Make sure that you get the study guide for the version you are taking as well, because they are different. Also it can be proctored by our education offices overseas, and ATI Testing doesn't even make us pay for the test (which is $100). Pretty good deal. Just email ATI Testing when you are getting ready to take it (atitesting.com).
Most states will give you in-state tuition because you are AD military and you will be on orders once you are accepted into the program, so when you look at prices, look at the in-state tuition rate (FL schools definitely do this). Here is the site with the list of all of the schools that you can apply to (all of these have ROTC detachments). Not every school on here has a nursing program, but every one that does on this site has the proper accreditations. Makes it a lot easier than trying to cross reference every school against NLNAC and CCNE accreditation websites.
If you don't already have a Bachelor's, you want an undergraduate traditional BSN program (can't exceed two years though). Accelerated are for those who already have degrees (in case you weren't sure).
Please don't hesitate to ask if you have any other questions, I am on here almost daily. Good luck and stay safe over there!