Global food security is at risk due to climate change. Soil fertility loss is among the impacts of climate change which reduces the productivity of rice–wheat cropping systems. This study investigated the effects of varying nitrogen levels and transplanting/sowing dates on the grain yield (GY) and biological yield (BY) of rice and wheat cultivars over two growing seasons (2017–2019). Additionally, the impact of climate change on the productivity of both crops was tested under a 1.5 °C temperature increase and 510 ppm CO2 concentration while nitrogen fertilization and sowing window adjustments were evaluated as adaptation options using the DSSAT and APSIM models. Results indicated that the application of 120 kg N ha−1 significantly enhanced both GY and BY in all rice cultivars. The highest wheat yields were obtained with 140 kg N ha−1 for all cultivars. Rice transplanting on the 1st of July and wheat sowing on the 15th of November showed the best yields. The statistical indices of the model’s forecast results were satisfactory for rice (R2 = 0.83–0.85, root mean square error (RMSE) = 341–441, model efficiency (EF) = 0.82–0.89) and wheat (R2 = 0.84–0.89, RMSE = 213–303, EF = 0.88–0.91). Both models predicted yield loss in wheat (20–25%) and rice (28–30%) under a climate change scenario. The models also predicted that increased nitrogen application and earlier planting would be necessary to reduce the impacts of climate change on the productivity of both crops.