Short answer: Mark the job as urgent, assign it to the nearest qualified person, make clear what moves down their list, and inform anyone affected by the change.
Triage first
Is it really urgent? Safety and work-stopping issues are. Most other things can wait a few hours.
Rebalance openly
Tell the person which task moves and update its deadline.
Learn from it
If the same urgent jobs keep happening, add preventive tasks.
Frequently asked questions
How do I stop everything being urgent?
Define urgent clearly and stick to it.
Try it with your team
Agree what counts as urgent with your team. Please Do It is free for 30 days, with no credit card required.