Neural implicit representations have emerged as a promising solution for providing dense geometry in Simultaneous Localization and Mapping (SLAM). However, existing methods in this direction fall short in terms of global consistency and low latency. This paper presents NGEL-SLAM to tackle the above challenges. To ensure global consistency, our system leverages a traditional feature-based tracking module that incorporates loop closure. Additionally, we maintain a global consistent map by representing the scene using multiple neural implicit fields, enabling quick adjustment to the loop closure. Moreover, our system allows for fast convergence through the use of octree-based implicit representations. The combination of rapid response to loop closure and fast convergence makes our system a truly low-latency system that achieves global consistency. Our system enables rendering high-fidelity RGB-D images, along with extracting dense and complete surfaces. Experiments on both synthetic and real-world datasets suggest that our system achieves state-of-the-art tracking and mapping accuracy while maintaining low latency.