The purpose of this paper is to analyze the distribution distance between random vectors derived from the magnitude of the analytic wavelet transform of the squared envelopes of Gaussian processes and their large-scale limits. When the Hurst index of the underlying Gaussian process falls below 1/2, the large-scale limit takes the form of a combination of the second and fourth Wiener chaos. In this case, we employ the non-Stein approach to determine the convergence rate in the Kolmogorov metric. In scenarios where the Hurst index of the underlying Gaussian process exceeds 1/2, the large-scale limit is a correlated chi-distributed random process with two degrees of freedom. The convergence rate in the Wasserstein metric is derived through the application of Malliavin calculus and multidimensional Stein's method. Although the large-scale limits are the same for Hurst indices within the interval (1/2,1), notable differences in the upper bounds of convergence rates are observed among Hurst indices falling within the subintervals (1/2,3/4) and (3/4,1).