This paper investigates a wireless covert communication system in which a transmitter aims to deliver messages to an intended receiver without being detected by a warden. Due to the absence of a direct link from the transmitter to the receiver, we propose a dual unmanned aerial vehicle (UAV) architecture consisting of a UAV-mounted reconfigurable intelligent surface (UAV-RIS) that establishes a controllable reflecting path and a dedicated UAV jammer that radiates artificial noise (AN) to confuse the detection of the warden. We formulate an optimization problem to maximize the covert rate of the legitimate transceiver pair, subject to a low outage probability constraint for the desired receiver and a high detection error probability (DEP) constraint for the warden. We establish an efficient block coordinate descent (BCD) optimization framework to iteratively design the transmit power, jamming power, and RIS phase shifts, where closed-form expressions for the first two power variables are derived in the process. Simulation results are provided to demonstrate the effectiveness of our proposed cooperative transmitting and jamming scheme via dual UAVs in achieving reliable and covert communication systems, particularly in highly obstructed scenarios.