If we assume that each task delivery is independent of eachother, and the process does not change a lot over time (stationary), we can treat this as a standard regression problem.
Since this is about deadlines, we expect that there might be variations over time, or patterns of delay across the seasons of the year or week. So time-based features might look something like:
We also expect that the size of a delay might depend on the size of the task. So if you have the start date, or an estimate on number of days, definitely include that. If people can have multiple tasks at the same time, include that also.
And we expect that delays may depend on the person who performs the task, and who created the task.
Use Exploratory Data Analysis and your knowledge about the processes that created to find more of these possible relationships. Use scatterplots of each feature against the target
days_delayed (negative=before time, 0=on time).
One can start with a strong non-linear model like RandomForest. This can give estimates which can be scored (by mean squared error for example), and indicate whether your features are predictive or not.
To get probability intervals, you can use a Bayesian model such as Bayesian Ridge Regression. This is a linear model, so may have to spend more time on feature engineering to make the relationships between feature and target (roughly) linear.