Cross-lingual biomedical entity linking (BEL) maps mentions in any language to unique identifiers in a biomedical knowledge base (KB), supporting clinical and biomedical NLP applications. However, expert-annotated training data for BEL are costly, especially for low-resource languages. Moreover, many cross-lingual BEL systems rely on SapBERT-based retrievers trained on predominantly English aliases in the KB, leading to poor generalization to unseen non-English mentions and limited context-aware disambiguation. We propose BioELX, a two-stage cross-lingual BEL framework that requires no task-specific annotated training corpora. In Stage~1, we enrich SapBERT training with Wikidata-derived multilingual aliases and use the resulting retriever to improve cross-lingual candidate retrieval. In Stage~2, we perform context-aware disambiguation with a pre-trained LLM ranker that jointly considers the mention context and candidate, eliminating the need for supervised training. Experiments on five benchmarks (XL-BEL, EMEA, Patent, WikiMed-DE, and MedMentions) show that BioELX achieves new state-of-the-art performance. It improves average Recall@1 on XL-BEL by +19.2, with especially large gains for low-resource languages, e.g., +21.6 on Turkish, +22.1 on Korean, +30.8 on Thai, and delivers consistent improvements on EMEA (+6.2), Patent (+5.4), and WikiMed-DE (+12.8). Code and resources will be released upon publication.