The integration of Large Language Models (LLMs) into autonomous driving systems demonstrates strong common sense and reasoning abilities, effectively addressing the pitfalls of purely data-driven methods. Current LLM-based agents require lengthy inference times and face challenges in interacting with real-time autonomous driving environments. A key open question is whether we can effectively leverage the knowledge from LLMs to train an efficient and robust Reinforcement Learning (RL) agent. This paper introduces RAPID, a novel \underline{\textbf{R}}obust \underline{\textbf{A}}daptive \underline{\textbf{P}}olicy \underline{\textbf{I}}nfusion and \underline{\textbf{D}}istillation framework, which trains specialized mix-of-policy RL agents using data synthesized by an LLM-based driving agent and online adaptation. RAPID features three key designs: 1) utilization of offline data collected from an LLM agent to distil expert knowledge into RL policies for faster real-time inference; 2) introduction of robust distillation in RL to inherit both performance and robustness from LLM-based teacher; and 3) employment of a mix-of-policy approach for joint decision decoding with a policy adapter. Through fine-tuning via online environment interaction, RAPID reduces the forgetting of LLM knowledge while maintaining adaptability to different tasks. Extensive experiments demonstrate RAPID's capability to effectively integrate LLM knowledge into scaled-down RL policies in an efficient, adaptable, and robust way. Code and checkpoints will be made publicly available upon acceptance.