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, $f\in\mathcal{B}^s_{π,r}$, and for a variety of $L^p$ error measures. We consider error distributions with Long-Range-Dependence parameter $α,0<α\leq1$; 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 $f-\int f$. The theory is illustrated with some numerical examples.