We investigate function estimation in nonparametric regression models with random design and heteroscedastic correlated noise. Adaptive properties of warped wavelet nonlinear approximations are studied over a wide range of Besov scales, $f\in\mathcal{B}_{\pi,r}^{s}$ , and for a variety of Lp error measures. We consider error distributions with Long-Range-Dependence parameter α, 0<α≤1; heteroscedasticity is modeled with a design dependent function σ. We prescribe a tuning paradigm, under which warped wavelet estimation achieves partial or full adaptivity results with the rates that are shown to be the minimax rates of convergence. For p>2, it is seen that there are three rate phases, namely the dense, sparse and long range dependence phase, depending on the relative values of s, p, π and α. Furthermore, we show that long range dependence does not come into play for shape estimation f−∫f. The theory is illustrated with some numerical examples.