As a 43yo developer, my advice is to push back on this. Make it clear from the first moment that it's their problem, not yours.
You can do this in multiple ways, like saying "You seem to have gotten yourself into trouble. Maybe next time, come to talk to us first so you don't make this huge mistake again". Don't give in to their smooth management talk, keep your ground.
They will try to let you solve the problem, but that's impossible. Let them solve their own mistake. If it's impossible to solve, well, it's a hard lesson for them.
At least in Europe, there is a huge shortage of developers. That means your employer needs you more than you need your employer (plenty of other employers out there). So stand your ground and make sure the problem stays on their end, not yours. And keep reminding them that they have to come to you for proper estimates.
You can already do this as a single developer. "This needs to be finished by Monday, can you make sure it is done?", "I'll do my best" (never promise!)
When they do come to you for estimates, make sure you know the difference between estimates and deadlines, and assume they don't know the difference. Make sure nobody confuses your estimates for deadlines. Better yet, never let them see your estimates, and only give the deadlines.