Large Language Models (LLMs) enable various applications on edge devices such as smartphones, wearables, and embodied robots. However, their deployment often depends on expensive cloud-based APIs, creating high operational costs, which limit access for smaller organizations and raise sustainability concerns. Certain LLMs can be deployed on-device, offering a cost-effective solution with reduced latency and improved privacy. Yet, limited computing resources constrain the size and accuracy of models that can be deployed, necessitating a collaborative design between edge and cloud. We propose a fast and cost-effective speculative edge-cloud decoding framework with a large target model on the server and a small draft model on the device. By introducing early exits in the target model, tokens are generated mid-verification, allowing the client to preemptively draft subsequent tokens before final verification, thus utilizing idle time and enhancing parallelism between edge and cloud. Using an NVIDIA Jetson Nano (client) and an A100 GPU (server) with Vicuna-68M (draft) and Llama2-7B (target) models, our method achieves up to a 35% reduction in latency compared to cloud-based autoregressive decoding, with an additional 11% improvement from preemptive drafting. To demonstrate real-world applicability, we deploy our method on the Unitree Go2 quadruped robot using Vision-Language Model (VLM) based control, achieving a 21% speedup over traditional cloud-based autoregressive decoding. These results demonstrate the potential of our framework for real-time LLM and VLM applications on resource-constrained edge devices.