The surprising thing about project managers who use the phrase "when-will-it-be-done" far too much is that they're often willing to help you with getting resources and getting help/cooperation "across silos" if you ask them in the right way.

They want the project to progress. Desperately.

From their point of view they're often powerless people surrounded by line workers that don't want to interact with them, yet they're somehow responsible for the success of the project. They're acutely aware that the meetings are often pointless for workers, deep inside, they know that coloring red/yellow/green rectangles on a gantt chart is of little value.

You can use that to your advantage if you make a request and frame it such that they can advance your progress concretely by helping you in some way that isn't possible for you to do alone.