I think either are great options. Mech E is group 1 and Math is group 2. My best friend from USNA was a math major. I had several friends who were Mech E majors. My best friend attended NPS as a LT and earned their Masters in Operational Research. Remember that for service assignment, you can earn any assignment regardless of major. A Electrical Engineer could be doing the exact same job on a ship as an English major. In the fleet it doesn't really matter, the Navy and/or Marine Corps will train you for your job. Will math or Mech E track help in something along the Nuke pipeline, absolutely. Will it open doors for things related to shore tours/b billets, yes. It could open doors to certain jobs while on shore tour as test leads, acquisition projects, requirements management, test pilot, research positions at ONR or other places. The Navy does have positions called Engineering Duty Officers. Essentially once you earn your pin you can then transfer to EDO which would put you in a different career path and out of the operational pipeline. USNA holds lots of open houses during Plebe Year for you to explore majors and discuss options. Your upperclass are also great resources for information about majors also. There are tons of post grad opportunities in the Navy and Marine Corps. When its best to attend and where to go will depend on your career field in the Navy or Marine Corps. These are things that you will learn as a Mid and as a JO.
