Bridge to Manufacturing Jobs Act - Amends the National Institute of Standards and Technology Act to: (1) provide that themes for the competitive grant program to add capabilities to the Manufacturing Extension Partnership Program may be related to projects designed to facilitate the development and implementation of apprenticeship programs, internship programs, or student work-based learning programs to help serve the workforce needs of consortia of manufacturing companies; and (2) include among the selection criteria for grants under such program that a proposal will recruit a diverse manufacturing workforce, including through outreach to women and minorities.