Metastatic disease is the leading cause of cancer-related death, despite recent advances in therapeutic interventions. Prior modeling approaches have accounted for the adaptive immune system's role in combating tumors, which has led to the development of stochastic models that explain cancer immunoediting and tumor-immune co-evolution. However, cancer immune-mediated dormancy, wherein the adaptive immune system maintains a micrometastatic population by keeping its growth in check, remains poorly understood. Immune-mediated dormancy can significantly delay the emergence (and therefore detection) of metastasis. An improved quantitative understanding of this process will thereby improve our ability to identify and treat cancer during the micrometastatic period. Here, we introduce a generalized stochastic model that incorporates the dynamic effects of immunomodulation within the tumor microenvironment on T cell-mediated cancer killing. This broad class of nonlinear birth-death model can account for a variety of cytotoxic T cell immunosuppressive effects, including regulatory T cells, cancer-associated fibroblasts, and myeloid-derived suppressor cells. We develop analytic expressions for the likelihood and mean time of immune escape. We also develop a method for identifying a corresponding diffusion approximation applicable to estimating population dynamics across a wide range of nonlinear birth-death processes. Lastly, we apply our model to estimate the nature and extent of immunomodulation that best explains the timing of disease recurrence in bladder and breast cancer patients. Our findings quantify the effects that stochastic tumor-immune interaction dynamics can play in the timing and likelihood of disease progression. Our analytical approximations provide a method of studying population escape in other ecological contexts involving nonlinear transition rates.