Amidst the rapid advances of large language models (LLMs), most LLMs still struggle with mixed-language inputs, limited Code-switching (CSW) datasets, and evaluation biases, which hinder their deployment in multilingual societies. This survey provides the first comprehensive analysis of CSW-aware LLM research, reviewing 327 studies spanning five research areas, 15+ NLP tasks, 30+ datasets, and 80+ languages. We classify recent advances by architecture, training strategy, and evaluation methodology, outlining how LLMs have reshaped CSW modelling and what challenges persist. The paper concludes with a roadmap emphasizing the need for inclusive datasets, fair evaluation, and linguistically grounded models to achieve truly multilingual intelligence. A curated collection of all resources is maintained at https://github.com/lingo-iitgn/awesome-code-mixing/.