Accurate prediction of injection profiles is a critical aspect of linking injector operation with engine performance and emissions. However, highly resolved injector simulations can take one to two weeks of wall-clock time, which is incompatible with engine design cycles with desired turnaround times of less than a day. Hence, it is important to reduce the time-to-solution of the internal flow simulations by several orders of magnitude to make it compatible with engine simulations. This work demonstrates a data-driven approach for tackling the computational overhead of injector simulations, via emulation of transient injection profiles for a side-oriented, single-hole diesel injector using a Bayesian machine-learning framework. An interpretable Bayesian learning strategy was employed to understand the effect of design parameters on the total void fraction field. Then, autoencoders are utilized for efficient dimensionality reduction of the flowfields. Gaussian process models are finally used to predict the spatiotemporal void fraction field at the injector exit for unknown operating conditions. The Gaussian process models produce principled uncertainty estimates associated with the emulated flowfields, providing the user with information about where the predictions can be trusted in the design space. The Bayesian predictions are compared with those obtained from a deep neural network, transfer-learned from static needle simulations discussed in previous work. The emulation framework can predict the void fraction at the exit of the orifice within a few seconds, achieving a speedup factor of ~38 million over the traditional CFD approach of generating transient injection maps.