Veteran Urgent Access to Mental Healthcare Act This bill directs the Department of Veterans Affairs (VA) to furnish to former members of the Armed Forces or the reserve components: (1) an initial mental health assessment; and (2) the mental health care services required to treat the member's urgent mental health care needs, including risk of suicide or harming others. A former member of the Armed Forces is an individual who: served in the active military, naval, or air service, was discharged or released under a condition that is not honorable (except a dishonorable or bad conduct discharge), has applied for a character of service determination that has not yet been made, and is not otherwise eligible to enroll in the VA health care system by reason of such discharge or release; or while serving in the Armed Forces, was deployed in a theater of combat operations or an area at a time during which hostilities occurred in that area, participated in or experienced such combat operations or hostilities, or was the victim of a physical assault of a sexual nature, battery of a sexual nature, or sexual harassment. The VA may provide such mental health care services at a non-VA facility under specifed circumstances.