The Coronavirus (COVID-19) pandemic has led to a rapidly growing 'infodemic' of health information online. This has motivated the need for accurate semantic search and retrieval of reliable COVID-19 information across millions of documents, in multiple languages. To address this challenge, this paper proposes a novel high precision and high recall neural Multistage BiCross encoder approach. It is a sequential three-stage ranking pipeline which uses the Okapi BM25 retrieval algorithm and transformer-based bi-encoder and cross-encoder to effectively rank the documents with respect to the given query. We present experimental results from our participation in the Multilingual Information Access (MLIA) shared task on COVID-19 multilingual semantic search. The independently evaluated MLIA results validate our approach and demonstrate that it outperforms other state-of-the-art approaches according to nearly all evaluation metrics in cases of both monolingual and bilingual runs.