The Navy, Nuke program in particular, is cracking down heavily on alcohol related incidents (DUI, underage drinking, etc.) so you're going to have a hell of a ride at best trying to get waivers for a DUI to become a Nuke. If you really want the job however, go for it. The worst that can happen is you go through MEPS and they say no, you get 2 free meals and a hotel stay out of it and another life experience under your belt. Don't take the decision lightly either, the pipeline sucks hardcore (I'm 7 months into my 6 month prototype, entered the program with high interest in learning about splitting atoms and have lost that interest substantially, though I still enjoy my job and look forward to what will come of my time in); think about it hard, figure out if you are indeed ready to go through 2+ years of training and then 4 years in the fleet (with minimal shore time) and if you're willing to take a shot to see if they're open to giving you a chance then go for it, it's not really a huge deal if you get declined - chalk it up to a lesson learned and move on.