Recently, time-varying linear equality and inequality equations (TVLEIE) is becoming increasingly crucial for solving problems in various fields. Zeroing neural networks (ZNNs) can also be employed to address the TVLEIE. Generally, the design convergent parameters (DCPs) of ZNN schemes affect the convergent speed. Since the previous fixed-parameter ZNNs (FPZNNs) use fixed parameters, they are not suitable for real-world applications where parameters vary over time. Taking this into account, the varying-parameter ZNNs (VPZNNs) were introduced in this field. Although the VPZNNs surpass the FPZNNs, their DCPs typically continue to increase over time and can even become excessively large in the end. But extremely large parameters are unsuitable. Moreover, the increasing parameters can lead to wasted computing resources, even when the VPZNNs become convergent. According to these considerations, we put forward a novel varying-parameter ZNN (NVPZNN) scheme with prescribed-time (PT) convergence to address the TVLEIE. NVPZNN has the capability to adjust its DCPs to progressively converge to a constant once it achieves convergence within the prescribed time. Subsequently, the global and PT convergence of NVPZNNs and their upper bounds as well as stability are theoretically analyzed. In comparison to other ZNN schemes utilizing common activation functions (AFs), the NVPZNN schemes own faster convergent rate, shorter convergent time and superior stability. Numerical experiments are conducted to validate the effectiveness and advantages of the NVPZNN schemes. Moreover, the successful application of NVPZNN in UR5 Manipulator shows its reliability and industrial application value.