The task of building a natural language interface to a database, known as NLIDB, has recently gained significant attention from both the database and Natural Language Processing (NLP) communities. With the proliferation of geospatial datasets driven by the rapid emergence of location-aware sensors, geospatial databases play a vital role in supporting geospatial applications. However, querying geospatial and temporal databases differs substantially from querying traditional relational databases due to the presence of geospatial topological operators and temporal operators. To bridge the gap between geospatial query languages and non-expert users, the geospatial research community has increasingly focused on developing NLIDBs for geospatial databases. Yet, existing research remains fragmented across systems, datasets, and methodological choices, making it difficult to clearly understand the landscape of existing methods, their strengths and weaknesses, and opportunities for future research. Existing surveys on NLIDBs focus on general-purpose database systems and do not treat geospatial and temporal databases as primary focus for analysis. To address this gap, this paper presents a comprehensive survey of studies on NLIDBs for geospatial and temporal databases. Specifically, we provide a detailed overview of datasets, evaluation metrics, and the taxonomy of the methods for geospatial and temporal NLIDBs, as well as a comparative analysis of the existing methods. Our survey reveals recurring trends in existing methods, substantial variation in datasets and evaluation practices, and several open challenges that continue to hinder progress in this area. Based on these findings, we identify promising directions for future research to advance natural language interfaces to geospatial and temporal databases.