Large language models (LLMs) are increasingly used to provide health advice, yet evidence on how their accuracy varies across languages, topics and information sources remains limited. We assess how linguistic and contextual factors affect the accuracy of AI-based health-claim verification. We evaluated seven widely used LLMs on two datasets: (i) 1,975 legally authorised nutrition and health claims from UK and EU regulatory registers translated into 21 languages; and (ii) 9,088 journalist-vetted public-health claims from the PUBHEALTH corpus spanning COVID-19, abortion, politics and general health, drawn from government advisories, scientific abstracts and media sources. Models classified each claim as supported or unsupported using majority voting across repeated runs. Accuracy was analysed by language, topic, source and model. Accuracy on authorised claims was highest in English and closely related European languages and declined in several widely spoken non-European languages, decreasing with syntactic distance from English. On real-world public-health claims, accuracy was substantially lower and varied systematically by topic and source. Models performed best on COVID-19 and government-attributed claims and worst on general health and scientific abstracts. High performance on English, canonical health claims masks substantial context-dependent gaps. Differences in training data exposure, editorial framing and topic-specific tuning likely contribute to these disparities, which are comparable in magnitude to cross-language differences. LLM accuracy in health-claim verification depends strongly on language, topic and information source. English-language performance does not reliably generalise across contexts, underscoring the need for multilingual, domain-specific evaluation before deployment in public-health communication.