The evolution of traditional electricity distribution infrastructures towards Smart Grid networks has generated the need to carry out new research. There are many fields that have attracted the attention of researchers, among which is the improvement of the performance of the so-called Neighborhood Area Networks (NAN). In this sense, and given the critical nature of some of the data transmitted by these networks, maintaining an adequate quality of service (QoS) is absolutely necessary. In emergency situations, this need becomes even more evident. This article presents a congestion control mechanism, whose parameters are modified according to the network state of emergency. The mechanism also applies a multi-channel allocation technique, together with a differentiation in the QoS offered to the different data flows according to their relevance. These proposals have been evaluated in the context of a wireless mesh networks (WMN) made up by a set of smart meter devices, where various smart grids (SG) applications are sending their data traffics. Each SG application must meet its unique quality of service (QoS) requirements, such as reliability and delay. To evaluate the proposals, some NAN scenarios have been built by using the ns-3 simulator and its 802.11s basic model, which was modified to implement the proposed techniques. Compared with the basic Hybrid Wireless Mesh Protocol (HWMP), Emergency Aware Congestion Control proposal (EA-HWMP), shows significant improvements in terms of packet delivery ratio, network throughput and transit time.