This paper investigates the asymptotic properties of the Kernel estimators for the conditional mean and variance functions in a time-varying nonlinear autoregressive locally stationary process with ARCH errors (tvNAR) of order d > 1. We establish the uniform almost sure consistency for these estimators with rate under assumptions of local stationarity and local ergodicity. Special attention is also paid to the Kernel density estimation of the tvNAR process. The results are derived without assuming a specific type of mixing conditions or a particular physical dependence measure on the data. Consequently, our results are applicable to a wide range of dependent processes, including those satisfying strong mixing conditions.