Recent integration of Natural Language Processing (NLP) and multimodal models has advanced the field of sports analytics. This survey presents a comprehensive review of the datasets and applications driving these innovations post-2020. We overviewed and categorized datasets into three primary types: language-based, multimodal, and convertible datasets. Language-based and multimodal datasets are for tasks involving text or multimodality (e.g., text, video, audio), respectively. Convertible datasets, initially single-modal (video), can be enriched with additional annotations, such as explanations of actions and video descriptions, to become multimodal, offering future potential for richer and more diverse applications. Our study highlights the contributions of these datasets to various applications, from improving fan experiences to supporting tactical analysis and medical diagnostics. We also discuss the challenges and future directions in dataset development, emphasizing the need for diverse, high-quality data to support real-time processing and personalized user experiences. This survey provides a foundational resource for researchers and practitioners aiming to leverage NLP and multimodal models in sports, offering insights into current trends and future opportunities in the field.