Unlawful Gun Buyer Alert Act This bill establishes a notification requirement following a firearms-related background check that results in a denial. Specifically, the bill requires the national instant criminal background check system to notify federal, state, and local law enforcement agencies if a firearms-related background check determines that an individual is ineligible to receive or possess a firearm. The notification requirement applies even if more than three business days have elapsed since the background check was initiated, and a firearm has already been transferred to the individual.