In this paper, we study the parameterized complexity of several variants of scheduling with precedence constraints between jobs. Namely, we consider the single machine setting with delay values on top of the precedence constraints. Such scheduling problems are related to several decades-old problems with open parameterized complexity status, notably Shuffle Product and Directed Bandwidth. We obtain XNLP-completeness results for both problems, and derive implications to scheduling with minimum (resp. maximum) delays parameterized by the width of the directed acyclic graph giving the precedence constraints, and/or by the maximum delay value in the input. Regarding Directed Bandwidth, we also settle the case of trees by showing XNLP-completeness parameterized by the target value. Beyond these results, we believe that Shuffle Product is an unusual and promising addition to the list of XNLP-complete problems.