Veterans Need Timely Access to Care Act - Directs the Secretary of Veterans Affairs (VA) to enter into contracts with such non-VA facilities as may be necessary to furnish hospital care or medical services to any veteran enrolled in the VA's patient enrollment system who: (1) has waited 7 days or longer for an appointment for primary care, or 14 days or longer for an appointment for speciality care, in a VA facility; and (2) elects to receive such care in a non-VA facility.