Suppose you have a prediction model for a quantity Y: Y=f(X), where X is the vector of the influence quantities. You first need to quantify the uncertainty of all influence quantities X, and then calculate the uncertainty of Y based on the law of propagation of uncertainty.

In my view, the question may have a two-pronged response. It can be about a time series forecast (I tend to call a forecast) or a regression based forecast (I casually call a prediction). The calculation of confidence limits of a predicted value in regression is widely available and is based on the variance of residuals - the differences between the observed values and the predicted values. This can be considered, using your terminology, as "uncertainty of the predicted value". Usually, we do not predict outside the limits of our independent variable. But with time series, this is a little more complicated. Our main purpose is to forecast outside the time period for which we have values. As we forecast values further and further away from the known time period, our confidence in the forecast further reduces, i.e. the uncertainty grows. To my knowledge, there is no single method for such calculations. In practice, I could not muster much confidence in such confidence intervals. A crude way to calculate such an interval is based on the standard deviation of the residuals for the known period. Then this can be adjusted by the forecast horizon. To directly answer your question as a pragmatist, yes, there are tools to "forecast the uncertainty in predictions", in my experience, with some 'uncertainty'! Hope this helps.

Yes, we can forecast uncertainty in predictions by quantifying and anticipating the degree of uncertainty through techniques like probabilistic forecasting, confidence intervals, and scenario analysis. These methods help account for model limitations, data inaccuracies, and unpredictable factors, providing a clearer picture of the range of possible outcomes.