Naturalization at Training Sites Act of 2017 or the NATS Act This bill directs the Department of Defense, and the Department of Homeland Security with respect to the Coast Guard, to establish a naturalization office with trained personnel at each initial military training site of the Armed Forces. The military department concerned shall: (1) identify members of the Armed Forces who are not U.S. citizens and inform them of the availability of naturalization services at such offices; and (2) pending establishment of a naturalization office at all initial military training sites, assign new non-citizen members of the Armed Forces to an initial military training site that has such an office.