This article is devoted to addressing the cloud control of connected vehicles, specifically focusing on analyzing the effect of bidirectional communication-induced delays. To mitigate the adverse effects of such delays, a novel predictor-observer structured controller is proposed which compensates for both measurable output delays and unmeasurable, yet bounded, input delays simultaneously. The study begins by novelly constructing an equivalent delay-free interconnected system model that incorporates the predictorobserver controller, considering certain delay boundaries and model uncertainties. Subsequently, a stability analysis is conducted to assess the system's robustness under these conditions. Next, the connected vehicle lateral control scenario which contains high-fidelity vehicle dynamic model is built. The results demonstrate the controller's ability to accurately predict the system states, even under time-varying bidirectional delays. Finally, the proposed method is deployed in a real connected vehicle lateral control system. Comparative tests with a conventional linear feedback controller showcase significantly improved control performance under dominant bidirectional delay conditions, affirming the superiority of the proposed method against the delay.