In this work, we present an algorithm for phaseless reconstruction from magnitude-only wavelet coefficients. The method relies on an explicit relation between the log-magnitude and phase gradients of analytic wavelet transforms and an extension of the Phase-Gradient Heap Integration (PGHI) algorithm recently introduced for Gabor phaseless reconstruction. This relation is exact for a certain family of mother wavelets including Cauchy wavelets of arbitrary order, but only holds approximately otherwise. The presented experiments show that, in practice, the proposed wavelet PGHI method provides competitive quality for various mother wavelets. Furthermore, wavelet PGHI is a non-iterative scheme and thus computational performance is significantly better than established alternate projection methods.