5G networks are considered potential enablers for many emerging edge applications, such as those related to autonomous vehicles, virtual and augmented reality, and online gaming. However, recent works have shown the cellular control plane is a potential bottleneck in enabling such applications - - control plane operations are slow, frequent, and can directly impact the delay experienced by end-user applications. Moreover, failures in the cellular control plane can significantly degrade application performance. In this paper, we consider the problem of enabling latency-sensitive and safety-critical edge applications on 5G networks. We identify fundamental control plane design challenges and posit enabling these applications requires re-thinking the cellular control plane. We propose a new edge-based cellular control plane, CellClone, which provides fast and fault-tolerant control plane processing. CellClone employs multiple active control plane clones at the network edge to mask control plane faults and speedup control processing. Central to its design is a custom quorum-based consistency protocol that provides state consistency with low latency. Testbed evaluations using real cellular traces show a median improvement of more than 3.8× in speeding up control plane operations with outright node failures and stragglers. These improvements translate into better application performance; with CellClone, autonomous cars and VR applications reduce missed application deadlines by more than 90%.