A new variable bandwidth multicarrier (VBMC) waveform was presented in [1] for communicating over wideband rapidly time-varying multi-scale multi-lag (MSML) channels. Perfect channel state information was assumed to be available at the receiver in [1]. In this work, we address the problem of channel estimation for VBMC based communications over wideband MSML channels. Using the Variational Bayesian (VB) inference framework, we estimate the channel from short preamble and postamble waveforms that are primarily used for timing and carrier frequency synchronization, and then decode the data symbols in VBMC communications. We also derive the Bayesian Cramér-Rao bound (BCRB) of the channel estimate as a benchmark for assessing the normalized mean squared error (NMSE) performance of the estimators. We numerically illustrate the efficacy of our approach in the context of underwater acoustic channel estimation.