The chemical processing of mixed textile waste remains a major challenge due to the heterogeneous composition of the materials. This study investigated a sequential chemical recycling approach for polycotton textile waste consisting of a 50:50 blend of polyester and cotton (cellulose) fibers. The first stage involved alkaline solvolysis in a glycol-based reaction medium aimed at selective depolymerization of the polyester fraction. The experimental study was based on a systematic experimental design to investigate the influence of key process parameters on polyester depolymerization. The experiments were performed under varying conditions, including temperatures in the range of 160–180 °C, reaction times of 10–30 min, fiber-to-liquid ratios up to 20 wt.%, and alkaline catalyst concentrations between 0.5–2.5 wt.% KOH. Following the alkaline solvolysis, step the remaining cotton (cellulose)-rich fraction was subjected to acid solvolysis. Statistical evaluation of the data obtained, including analysis of variance, made it possible to identify the most significant factors affecting the process and to determine the optimal operating conditions for maximizing polyester conversion from textile mixtures and for converting cellulose into low-molecular-weight chemicals. Reaction products and structural changes were characterized using FTIR. In addition, the environmental performance of the proposed processing method was assessed using a life cycle assessment (LCA) implemented in OpenLCA software. The results highlight the potential of a sequential alkaline and acid solvolysis strategies for the utilization of polycotton textile waste within a circular economy.