We propose a tree-search based Bayesian approach to blind maximum likelihood sequence detection (MLSD) of convolutionally encoded data transmitted over a multipath Rayleigh fading channel. In deriving the path metric for searching the channel-code tree, the proposed algorithm incorporates a forgetting factor matched to the time variation of the channel to generate accurate estimates of the correlation across the transmitted and received data. In addition, an augmented metric is presented to address the challenge of unknown channel order in time-varying systems. Simulation results show that the proposed algorithm can achieve significant improvement in bit error rate over competing schemes, even when channel order information is unavailable at the receiver.