From the perspective of supply chain system, providing customers with flexible uncertain delivery time can improve the competitiveness of the supply chain and customer satisfaction on one hand, while challenging the efficient operation of the supply chain system on the other hand. To this end, this paper considers the characteristics of collaborative enterprises in the supply chain collaborative manufacturing mode, such as different production startup costs as well as the need for advance preparation and long procurement lead time. We construct a two-stage supply chain collaborative robust scheduling optimization model based on the box uncertainty set of the due dates. This model is designed to reduce decision conservatism and optimize supply chain system costs. The column-and-constraint generation algorithm (C&CG) is used to solve this model exactly, and the subproblem is transformed based on the convexity of the objective function in the uncertainty to construct a simplified dual problem. Numerical simulation examples analyze the impact of uncertain due date change on the two-stage robust scheduling strategy and compare the robust optimal cost of single-stage, two-stage, and hindsight under different real due date scenarios. Meanwhile, the real cost under the deviation between the estimated and the real due date intervals is compared and analyzed. The results indicate that the cost of the two-stage robust model is significantly better than the cost of single-stage robust model, and the cost is not far from the ex-post optimal cost. In addition, the cost of the two-stage robust model shows relatively small variance, and the two-stage robust solution is much less sensitive to the estimation error of the due date intervals than the single-stage robust one. Therefore, the two-stage robust optimization model can significantly improve the conservatism and inflexibility of the scheduling strategies and effectively respond to the needs of flexible uncertain due dates.