This paper addresses the distributed recursive filtering problem for a class of time-varying multi-rate nonlinear systems which are subject to fading channels, sensor resolutions and quantization effects. For the purpose of prolonging the transmission distance, the amplify-and-forward (AaF) relay is deployed in wireless communication channel. The iterative method is employed to convert the underlying multi-rate system into a single-rate one. Before entering communication networks to transmission, measurement outputs are quantized by using the logarithmic quantizer. Moreover, a new equivalent delay-free measurement sequence is defined by the measurement reorganization technology to tackle random transmission delays. Our mainly aim is to design a distributed recursive filter such that an upper bound for the filtering error covariance is determined and subsequently minimized by properly choosing the filtering gains. Next, the performance of the proposed filtering algorithm is analyzed through establishing a sufficient condition for the boundedness of the filtering error covariance. Finally, a numerical simulation is used to testify the applicability and effectiveness of the developed filtering algorithm.