We address the challenge of estimating the hyperuniformity exponent $\alpha$ of a spatial point process, given only one realization of it. Assuming that the structure factor $S$ of the point process follows a vanishing power law at the origin (the typical case of a hyperuniform point process), this exponent is defined as the slope near the origin of $\log S$. Our estimator is built upon the (expanding window) asymptotic variance of some wavelet transforms of the point process. By combining several scales and several wavelets, we develop a multi-scale, multi-taper estimator $\widehat{\alpha}$. We analyze its asymptotic behavior, proving its consistency under various settings, and enabling the construction of asymptotic confidence intervals for $\alpha$ when $\alpha < d$ and under Brillinger mixing. This construction is derived from a multivariate central limit theorem where the normalisations are non-standard and vary among the components. We also present a non-asymptotic deviation inequality providing insights into the influence of tapers on the bias-variance trade-off of $\widehat{\alpha}$. Finally, we investigate the performance of $\widehat{\alpha}$ through simulations, and we apply our method to the analysis of hyperuniformity in a real dataset of marine algae.