Next-generation wireless networks are progressing beyond conventional connectivity to incorporate emerging sensing and computing capabilities. This convergence gives rise to integrated systems that enable not only uninterrupted communication, but also environmental awareness, intelligent decision-making, and novel applications that take advantage of these combined features. At the same time, this integration brings substantial security challenges. As computing, sensing, and communication become more tightly intertwined, the overall complexity of the system increases, creating new vulnerabilities and expanding the attack surface. The widespread deployment of data-heavy artificial intelligence applications further amplifies concerns regarding data security and privacy. This paper presents a comprehensive survey of security and privacy threats, along with potential countermeasures, in integrated wireless systems. We first review physical-layer security techniques for communication networks, and then investigate the security and privacy implications of semantic and pragmatic communications and their associated cross-layer design methodologies. For sensing functionalities, we pinpoint security and privacy risks at the levels of signal sources, propagation channels, and sensing targets, and summarize state-of-the-art defense strategies for each. The growing computational requirements of these applications drive the need for distributed computing over the network, which introduces additional risks such as data leakage, weak authentication, and multiple points of failure. We subsequently discuss secure coded computing approaches that can help overcome several of these challenges. Finally, we introduce unified security frameworks tailored to integrated communication-sensing-computing architectures, offering an end-to-end perspective on protecting future wireless systems.