Large Language Models (LLMs) have significantly propelled the advancement of edge intelligence and have been widely deployed across various scenarios, including autonomous driving, industrial inspection, and personalized IoT services. However, the collaborative adaptation of LLMs on edge devices continues to face formidable challenges due to strict data privacy constraints, highly heterogeneous computing and communication resources, and the non-independent and identically distributed (non-IID) nature of local data. Federated Fine-Tuning (FFT) enables the collaborative optimization of distributed models without exposing raw data. Yet, traditional synchronous aggregation suffers from a severe straggler effect, resulting in high system latency and low resource utilization. Existing asynchronous federated learning methods are predominantly designed for small-to-medium-scale models and struggle to address the specific challenges inherent in LLM fine-tuning namely, model drift caused by stale updates, aggravated client drift stemming from data heterogeneity, and aggregation fairness imbalance resulting from the dominance of fast clients. To address these issues, this paper proposes AlignFed, an asynchronous federated fine-tuning framework for LLMs tailored to heterogeneous edge environments. AlignFed employs a lightweight multi-stage semantic alignment mechanism comprising three core modules: version-aware update grouping, cross-version semantic alignment based on a mini-batch calibration set, and fairness-aware aggregation that integrates both update freshness and client participation frequency. This framework effectively mitigates cross-version model drift and client drift while enhancing aggregation fairness, thereby achieving stable and efficient asynchronous federated optimization in scenarios characterized by high heterogeneity and significant update staleness.