I’m running into a similar problem in my campaign... I want to create urgency around a possible apocalyptic event without using a hard timer. What I landed on is to give the PCs a look at a possible future. The next session will have them “transported” to a timeline where the worst has happened.

They’ll be shrouded by an illusion so no one recognizes them, but otherwise they’ll play their characters as-is, at current level. One of my recurring NPCs is 12 years old presently. In the future world, she’ll be in her early 30s and the leader of of a small cell of survivors. I plan to be intentionally vague about when exactly the world fell apart, while making it ugly enough to motivate them once they flash back to the present. It’s going to be tricky but I feel good about the idea.