In order to investigate the analysis and processing methods for nonstationary signals generated in bridge health monitoring systems, this study combines the advantages of complete ensemble empirical mode decomposition (CEEMD) and wavelet threshold denoising algorithms to construct the CEEMD–wavelet threshold denoising algorithm. The algorithm follows the following steps: first, add noise to the monitoring data and obtain all the mode components through empirical mode decomposition (EMD), denoise the mode components with noise using the wavelet threshold function to remove the noise components, select the optimal stratification for denoising the monitoring data of the Guozigou Bridge in Xinjiang in January 2023, determine the wavelet type and threshold selection criteria, and reconstruct the denoised intrinsic mode function (IMF) components to achieve accurate extraction of the effective signal. By referencing the deflection, temperature, and strain data of the Guozigou Bridge in Xinjiang in January 2023 and comparing the data cleaned by different mode decomposition and wavelet threshold denoising methods, the results show that compared with empirical mode decomposition (EMD)–wavelet threshold denoising and variational mode decomposition (VMD)–wavelet threshold denoising, the signal-to-noise ratios and root-mean-square errors of the four types of monitoring data obtained by the algorithm proposed in this study are the most ideal. Under the premise of minimizing reconstruction errors when processing a large amount of data, it has better convergence, verifying the practicality and reliability of the algorithm in the field of bridge health monitoring data cleaning and providing a certain reference value for further research in the field of signal processing. The computational method constructed in this study will provide theoretical support for data cleaning and analysis of nonstationary and nonlinear random signals, which is conducive to further promoting the improvement of bridge health monitoring systems.