I would suggest that when you do get your acceptance letter to go back to the recruiting (medical recruiting corps, not enlisted!) center. I started my application process when I moved to Maine in May (started classes in May) and I wasn't accepted until the middle of November and I swore in December. Having said that the application process took a good 4-6 months depending on how on top of things you are.
You want to get that application in quick because 1. they only have 1 board a year now where they only accept around 10-15 people (don't take my word on this though because it always changes based on the needs of the Army). 2. On top of the $10,000 bonus (paid in 2 $5k payments at the start and then at the end of your program), you get a $1,000 stipend for every month your in a nursing program. You want to get every bit of that if your set on the Army! Even though my check didn't come through until just this February since I swore in/signed my paperwork in November when I got accepted I got stipend back pay for Dec/Jan.
What they'll first have you do is take the ASVAB. This is because with the ANCP we get "enlisted" inactive ready reserve to be in the books to get our payments and then get commissioned as officers once we pass our NCLEX. Don't worry much of a stretch on this as its literally just like taking SAT level math and English. Those are the only sections that really matter in your composite score. I picked up an ASVAB for Dummies book and studied for a few weeks and got a 96 (you only need 50, but the higher the better your packet looks). Also try to get your GPA as high as possible before you go to board (most likely later this November). I think their looking for 3.3+ if possible. You'll also have to pass your physical at MEPS, which will most likely be the same time you take the ASVAB. Their pretty chill with officer recruits so don't worry about them intimidating the crap out of you. You'll be fine.
Paperwork...just make sure you have your social security card, birth certificate, all transcripts, and other material prepared. I think you also need to prepare three letters of recommendations. Make sure you have your professors write you out an actual letter instead of filling out the Army guideline (it just looks better and more professional) in your school's official letterhead. Then you'll have to fill out I think it was some 56 page monster security application (for all officers) to get your security clearance. You'll have to list former jobs, every address you've lived the past 7 years, and three friends they know you in case they ever need to ask questions about you. Don't sweat this either, nurses only need a secret clearance so their not going to throw the book at you. But down the line if you needed a top secret clearance based on your station they would do a more intensive background check.
So I heard that some people can get ALL this done in one month. It took me a good 3-4 months and I'd say I tried to be on top of things as most I could. But I was also taking a crapload of units so I wasn't always so free. I would recommend being really on top of things with your recruiter, if you show them your serious and don't lag with your stuff...trust me they'll pull their weight for you. Paperwork is a hassle because you're going to be filling out a lot of it, but if your heart is set on the Army trust me its worth it. They usually also want you to have your packet complete at least a month before the actual board date so like October if its in November. This is so they could send your packet and make the necessary changes if it happens to get bounced back. Or get any necessary waivers if needed.
I'd say it's competitive because they want high scores in both GPA and ASVAB because they have more applicants than spaces, but its also tough because you could get D/Qed during your physical if they find anything off. I hope I answered some of your questions and I'm sorry for ranting a bit. If you have any further questions feel free to ask! Good luck with your school applications and Army plans.