Theory and methodology for nonparametric regression have been particularly well developed in the case of additive homoscedastic Gaussian noise. Inspired by asymptotic equivalence theory, there have been ongoing efforts in recent years to construct explicit procedures that turn other function estimation problems into a standard nonparametric regression with Gaussian noise. Then in principle any good Gaussian nonparametric regression method can be used to solve those more complicated nonparametric models. In particular, Brown, Cai and Zhou [3] considered nonparametric regression in natural exponential families with a quadratic variance function. In this paper we extend the scope of Brown, Cai and Zhou [3] to general natural exponential families by introducing a new explicit procedure that is based on the variance stabilizing transformation. The new approach significantly reduces the bias of the inverse transformation and as a consequence it enables the method to be applicable to a wider class of exponential families. Combining this procedure with a wavelet block thresholding estimator for Gaussian nonparametric regression, we show that the resulting estimator enjoys a high degree of adaptivity and spatial adaptivity with near-optimal asymptotic performance over a broad range of Besov spaces.