The declining availability of natural sand resources and the significant carbon footprint associated with the extensive use of cement are posing severe limitations on the advancement and application of ultra-high-performance concrete (UHPC). In this study, waste tyre-derived recycled crumb rubber particles (CR) were employed to replace quartz sand, and an alkali-activated cementitious material was used to produce waste tyre-alkali-activated UHPC (T-UHPAC). The influence of different CR replacement ratios (0%, 5%, 20%, 35%, 50%) on the tensile and flexural performance of T-UHPAC was investigated, and a predictive model for the stress–strain response considering the CR replacement ratio was established. An optimization method for improving the tensile and flexural performance of T-UHPAC was proposed. The results indicate that the effect of rough-surfaced CR on the interfacial properties of concrete differs from that of smooth quartz sand. A CR replacement ratio exceeding 35% led to a reduction in both the tensile and flexural strengths of UHPAC, while a replacement ratio at or below 20% resulted in a superior tensile and flexural performance of T-UHPAC. The established predictive model for tensile performance accurately forecasts the stress–strain behaviour of T-UHPAC under varying CR replacement ratios, with the accuracy improving as the CR replacement ratio increases. By utilizing CR to replace quartz sand in proportions not exceeding 20%, the production of low-carbon UHPC with exceptional comprehensive mechanical properties is achievable. Moreover, the development of T-UHPAC through the comprehensive utilization of waste tyres presents a promising and innovative approach for the low-carbon and cost-effective production of UHPC, thereby facilitating the sustainable development of natural resources. This research represents a significant step towards the widespread adoption and application of UHPC and thus holds substantial importance.