No Nuclear Testing Without Approval Act This bill prohibits any explosive nuclear testing from being conducted by the United States after the enactment of this bill and no funds may be obligated or expended to conduct such testing unless (1) a foreign state conducts a nuclear test after that date, or (2) there is a technical need for such testing. The President must submit a notification to Congress that describes the proposed testing and states the reasons for conducting such testing. The bill sets forth procedures for a congressional joint resolution of approval for explosive nuclear testing for which there is a technical need.