Recently, federated large language models (LLMs) have drawn significant attention thanks to coupled capabilities of LLMs and federated learning (FL) that address privacy concerns in collaborative fine-tuning. However, due to large-scale parameters of LLMs, existing federated LLM fine-tuning frameworks incur significant challenges in resource-constrained clients characterized by heterogeneous computing capabilities and random wireless channels. To address this issue, we propose a joint client-specific pruning and bandwidth allocation (JCPBA) framework for federated LLMs to improve the fine-tuning efficiency over the wireless networks. Specifically, we formulate a fine-tuning latency minimization problem by jointly optimizing pruning rates and bandwidth allocations. Furthermore, we solve this optimization problem using a block coordinate descent method. Extensive experiments on the datasets of Yahoo Answers and GSM8K demonstrate that the proposed framework significantly reduces wall-clock fine-tuning time compared with state-of-the-art baselines and gains equal or lower test loss at the cost of lower computation and communication overhead.