I think you will have a very difficult time getting accepted to top tier programs such as U Michigan, MIT, Texas A&M, and UC Berkeley without a Bachelor's degree or research experience in the  field. You should keep in mind that most applicants possess not only a  strong academic/research background in nuclear physics, but are also  adept in computational methods. 
 
However, I know as a fact that CS undergraduate majors and non-majors in  general have gotten into the MS Nuclear Engineering programs at second  tier schools such as the University of Illinois at Urbana-Champaign and  Penn State University (where I got my undergrad and masters degree,  respectively). 
 
In my opinion, those schools offer an excellent nuclear engineering  program, and have courses specifically targeted towards non-majors  entering the graduate curriculum. The professors to talk to regarding  research in computational methods would be Dr. Uddin at UIUC and Dr.  Ivanov at PSU. You can email them explaining your educational background  and your interest in nuclear engineering, and what research they  envision you participating in as a Master's student.  
 
Also, your points of contacts are correct in that even if you are  admitted into the second-tier programs, you will have to take a handful  of junior/senior undergraduate level classes in order to meet the  requirements for the graduate level classes. As I said before, it will  be achievable but difficult! 
 
I also think you have adequate math experience in terms of an entering graduate student. I am sure if you do research in computational methods, there will be plenty more math e.g. numerical analysis to take. 
 
Anyway, in the mean time, I would highly suggest picking up and reading(cover to cover) either "Introduction to Nuclear Engineering" by John R. Lamarsh or "Fundamentals of Nuclear Science and Engineering" by Shultis and Faw.Those books will give you a great overview into every aspect of nuclear engineering. For computational methods in nuclear engineering, I would read "Computational Methods for Neutron Transport" by Lewis. That book will give you a very strong background on fundamental techniques for computational nuclear engineering. 
 
In addition, I would also read a few recent papers by professors(won't make sense to you unless you read Computational Methods for Neutron Transport) before applying, so that when you do apply, you can succinctly state your research area of interest. 
 
Alright. Good luck.