Embodied vision-based real-world systems, such as mobile robots, require a careful balance between energy consumption, compute latency, and safety constraints to optimize operation across dynamic tasks and contexts. As local computation tends to be restricted, offloading the computation, i.e., to a remote server, can save local resources while providing access to high-quality predictions from powerful and large models. However, the resulting communication and latency overhead has led to limited usability of cloud models in dynamic, safety-critical, real-time settings. To effectively address this trade-off, we introduce UniLCD, a novel hybrid inference framework for enabling flexible local-cloud collaboration. By efficiently optimizing a flexible routing module via reinforcement learning and a suitable multi-task objective, UniLCD is specifically designed to support the multiple constraints of safety-critical end-to-end mobile systems. We validate the proposed approach using a challenging, crowded navigation task requiring frequent and timely switching between local and cloud operations. UniLCD demonstrates improved overall performance and efficiency, by over 23% compared to state-of-the-art baselines based on various split computing and early exit strategies. Our code is available at https://unilcd.github.io/ .