When doing a project that requires design, you should plan the time required for design carefully and allow adequate time for the design to be completed.
You should also write a design brief describing what you expect the designers to deliver and get them to agree on a schedule and cost for this (even for an internal design department).
Make sure you also account for design review (by the senior designer), drafting, engineering review of the drafting, redrafting of changes and corrections, and final review and issue.
It is important to allow for all these things and get the design leader to agree on time allowances for this.
You may find that if a design leader has agreed to a brief (including expected delivery times) and signed it then they will be more likely to work on your designs instead of other departments or projects designs (who don’t have a clear agreement).
It is not enough to send the design lead a list of target dates or descriptions, you need to set it all out in a clear document and get them to agree to (and preferably sign) this.
Note that final design review and signoff can be a big bottle neck if there is only one designer authorised to do this.