Rapid developments of large language models have revolutionized many NLP tasks for English data. Unfortunately, the models and their evaluations for low-resource languages are being overlooked, especially for languages in South Asia. Although there are more than 650 languages in South Asia, many of them either have very limited computational resources or are missing from existing language models. Thus, a concrete question to be answered is: Can we assess the current stage and challenges to inform our NLP community and facilitate model developments for South Asian languages? In this survey, we have comprehensively examined current efforts and challenges of NLP models for South Asian languages by retrieving studies since 2020, with a focus on transformer-based models, such as BERT, T5, & GPT. We present advances and gaps across 3 essential aspects: data, models, & tasks, such as available data sources, fine-tuning strategies, & domain applications. Our findings highlight substantial issues, including missing data in critical domains (e.g., health), code-mixing, and lack of standardized evaluation benchmarks. Our survey aims to raise awareness within the NLP community for more targeted data curation, unify benchmarks tailored to cultural and linguistic nuances of South Asia, and encourage an equitable representation of South Asian languages. The complete list of resources is available at: https://github.com/trust-nlp/LM4SouthAsia-Survey.