You do not need 2 voltage regulators... just unplug the power supply, and plug the battery into the first LM2596.
and as per @Paul_B comment... you don't need to go through the power jack on the Arduino... you already have 5 volts, so just connect direct to the UNOs 5v line. Or if you do go through the power jack then you should adjust the output of the LM2596 to about 9v (the Arduino will reduce to 5v).