This paper considers nonparametric estimation and inference in first-order autoregressive (AR(1)) models with deterministically time-varying parameters. A key feature of the proposed approach is to allow for time-varying stationarity in some time periods, time-varying nonstationarity (i.e., unit root or local-to-unit root behavior) in other periods, and smooth transitions between the two. The estimation of the AR parameter at any time point is based on a local least squares regression method, where the relevant initial condition is endogenous. We obtain limit distributions for the AR parameter estimator and t-statistic at a given point τ in time when the parameter exhibits unit root, local-to-unity, or stationary/stationary-like behavior at time τ. These results are used to construct confidence intervals and median-unbiased interval estimators for the AR parameter at any specified point in time. The confidence intervals have correct uniform asymptotic coverage probability regardless of the time-varying stationarity/nonstationary behavior of the observations.