Online Bayesian learning-assisted channel state information (CSI) estimation schemes are conceived for single input single output (SISO) and multiple input multiple output (MIMO) orthogonal time frequency space (OTFS) modulated systems. To begin with, an end-to-end system model is derived in the delay-Doppler (DD)-domain, followed by an online CSI estimation (CE) framework for SISO-OTFS systems. Next, the sequential minimum mean square error (MMSE) estimator is derived for this model which utilizes expectation maximization (EM) based sparse Bayesian learning (SBL) for initialization of the online estimation procedure. Additionally, a low-complexity detection technique is developed for the system under consideration, which is accomplished via an analogous time-frequency (TF)-domain system model that leads to a block-diagonal TF-domain channel matrix. The paradigm designed for online CE is subsequently extended to MIMO-OTFS systems. The corresponding DD-domain CSI is shown to be simultaneously row and group sparse. Hence a novel EM-based row and group sparse Bayesian learning scheme is developed for determining the initialization parameters for the above online algorithm. As a further continuation, a low-complexity detector is also proposed for MIMO-OTFS systems based on an iterative block matrix inversion technique. Furthermore, time-recursive Bayesian Cramer-Rao lower bounds (BCRLBs) are derived to benchmark the MSE performance of the proposed schemes for both the systems. Finally, simulation results are presented to demonstrate the efficiency of the proposed online estimation techniques.