This paper focuses on the distributed control problem in a networked microgrid (NMG) with heterogeneous energy storage units (HESUs) in the environment considering multiple types of time delays, which include the state, input, and communication delays. To address this problem, a state feedback control (SFC) strategy based on nested predictor is proposed to mitigate the influence of multiple types of time delays. First, a distributed control method founded upon voltage observer is developed, which can realize proportional power distribution according to the state of charge (SOC) of the HESUs, while adjusting the average voltage of the point of common coupling (PCC) bus in the NMG to its rated value. Then, considering that there exists steady-state error resulting from the initial value of the observer and impact of time delays, an SFC strategy is proposed to further improve the robustness of the NMG against time delays. Finally, the experimental results demonstrate that the proposed distributed control method is capable of fully compensating for the state, input, and communication delay. Moreover, the NMG exhibits remarkable resistance to multiple types of time delays, which has higher reliability and robustness.