This article proposes a prescribed adaptive backstepping scheme with new filter-connected switched hysteretic quantizer (FCSHQ) for switched nonlinear systems with nonstrict-feedback structure and time-delay. The system model is subjected to unknown functions, unknown delays, and unknown Bouc-Wen hysteresis nonlinearity. The coexistence of quantized input and actuator hysteresis may deteriorate the shape of hysteresis loop and, consequently, fail to guarantee the stability. To deal with this issue, a new FCSHQ is introduced to smooth the input hysteresis. This adaptive filter also provides us a degree of freedom at choosing the desired communication rate. The repetitive differentiations of virtual control laws and existing a lot of learning parameters in the neural network (NN)-based controller may result in an algebraic loop problem and high computational time, especially in a nonstrict-feedback form. This challenge is eased by the key advantage of NNs' property where the upper bound of the weight vector is employed. Then, by an appropriate Lyapunov-Krasovskii functional, a common Lyapunov function is presented for all subsystems. It is shown that the proposed controller ensures the predefined output tracking accuracies and boundedness of the closed-loop signals under any arbitrary switching. Finally, the proposed control scheme is verified on a practical example where simulation results demonstrate the effectiveness of the proposed scheme.