There is already
introduced in the House that would allow more nuclear power plants to be built; it's HR 909, co-sponsored by 64 Republicans. The part you're looking for:
Subject to the requirements of this subtitle and in accordance with existing law, the Nuclear Regulatory Commission shall issue operating permits for 200 new commercial nuclear reactors, enough to triple current megawatt capacity, by 2040, if there are a sufficient number of qualified applicants.
The stumbling block is the bill's endorsement of waste storage at Yucca Mountain, something that Sen. Harry Reid, D-Nev., among other Democrats, has made politically impossible. But other Republican energy bills have deregulatory components. This doesn't.
Nothing in this title shall supersede, mitigate, detract from, or in any way decrease the Nuclear Regulatory Commission’s ability to maintain the highest possible levels of public health and safety standards, consistent with the provisions of the Atomic Energy Act of 1954. No authority granted by this title shall be executed in a manner that jeopardizes, minimizes, reduces, or lessens public health and safety standards.