Slightly more than two months for this to happen. Putin has been president for ~25 years, so initial estimate of an upper bound on the base rate is something like 1/(25*6), less than 1%. The inauguration after the next Russian election is 7 May, after deadline on this question.
Maybe the probability for falling out of power is higher now than it was 10 years ago thanks to the war, but it's unclear that it should be much higher.
Another way of estimating the probability, since I often mess up probability estimates with short time horizons: estimate a probability with a longer horizon, and divide. I think Putin will probably be in power in 5 years, and probably not in 10. Assuming conservatively that 5 years is the 50-50 point, and further assuming that dropping out of power is equally likely at any time (in particular no more likely right now than, say, next November), the probability for the next two months should be (0.5)/(5*6). Slightly more than 1%. I'll say 2% to account for the probability that I screwed up this forecast.
Slightly more than two weeks left; it doesn't look like the per-time probability is any higher than it was a couple weeks ago, so we're now <1%.