The "No Nuclear Testing Without Approval Act" seeks to establish a stringent requirement for Congressional approval before the United States can conduct any explosive nuclear testing. This legislation amends the Atomic Energy Defense Act, prohibiting such testing unless specific conditions are met and a joint resolution of approval is enacted into law. Explosive nuclear testing would only be permitted under two circumstances: if a foreign state first conducts an explosive nuclear test, or if there is a demonstrated "technical need" for such testing. A technical need is defined as a determination by specified officials that a test is necessary to resolve issues concerning the safety, reliability, performance, or military effectiveness of a nuclear weapon type. Before any testing can proceed, the President must submit a detailed notification to Congress at least 180 days prior to the proposed test date. This notification must include a description of the testing, the reasons for conducting it (including an assessment of alternatives for technical need, or a geopolitical explanation), estimated timelines, and costs. Congressional approval then requires the passage of a joint resolution. If testing is in response to a foreign test, the resolution requires a two-thirds affirmative vote of Senators. If testing is due to a technical need, the resolution follows specific expedited procedures in both the House and Senate, also culminating in a two-thirds affirmative vote in the Senate for passage.