A thin viscous accretion disc around a Kerr black hole, which is warped due to the Lense-Thirring (LT) effect, was shown to cause the spin axis of the black hole to precess and align with the outer disc. We calculate the total LT torque acting on the black hole, and compute the alignment and precession time-scales for both persistent and transient accretors. In our analysis, we consider the contribution of the inner disc, as it can stay misaligned with the black hole spin for a reasonable range of parameter values. We find that the alignment time-scale increases with a decrease in the Kerr parameter below a critical Kerr parameter value, contrary to earlier predictions. Besides, the time-scales are generally longer for transience than the time-scales calculated for persistent accretion. From our analysis of the transient case, we find that the black hole in the low mass X-ray binary (LMXB) 4U 1543-47 could be misaligned, whereas that in the LMXB XTE J1550-564 has aligned itself with the outer disc. The age of the LMXB H 1743-322 is estimated assuming a misaligned disc. We also find that the black hole in a typical Galactic LMXB can take a significantly longer time to align than what was estimated in the past. This may have an important implication on the measurement of black hole spin using the continuum X-ray spectral fitting method.
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