Guess I should chime in.
I feel your pain dude. I had two very severe panic attacks in the navy that lead me to getting discharged early. Some here want to call me a "sad panda", but that's not the case. I was diagnosed with anxiety, impulsive explosive disorder, depression, and "adjustment disorder" all in one go. I wasn't put on meds but I had to do a lot of counseling (still am).
Well after getting out, I applied to various plants, and a bunch were still interested. I did the testing, and it was scary/difficult explaining everything, but I was still hired. It may be tempting to try to cover it up or minimize it, but have faith in yourself. If you don't get hired in right now, keep taking your meds and doing what you have to do. I was prett surprised they still cleared me, but it was nice and helped me sleep at night knowing they knew everything already.
If you are still seeing a psychiatrist, tell him/her your goals of getting a nuke job, and see if they will give you a written recommendation, it will go pretty far for you.
Good luck. The user "graphic' here can help you out a lot as well. Id suggest sending him a pm and let him know what's going on. He's been invaluable to the site.
Edit
I didn't even answer your question! You will authorize them to "dig" as much as they want. To my knowledge they never looked at my medical records, maybe they did, maybe they didn't. I think they are more concerned with you being honest