Get to know the usual time schedules for producing plans, drawings, and designs in your organisation.
Estimates may be 2 weeks but people with experience in the organisation may know to double this when told 2 weeks by the design department.
Project managers should keep up with the designers for their project regularly to see what they are working on and what their priorities are.
Resources often get pulled onto other tasks/projects, so it important you stay informed.
Possibly escalate to the projects director if necessary.