A request can be urgent without making the recipient responsible for its urgency.

Deadlines have origins. The person receiving one did not usually create it. When a late request arrives dressed in alarm, the pressure gets transferred along with the task, so someone else now carries anxiety about a delay they had no part in. Stating the deadline plainly and owning where it came from keeps the urgency attached to the situation rather than to them. People help more readily when they are not also being blamed in advance.

When you send something urgent, say why it is late and what could still move. Ask what is possible rather than announcing what is required.