You need to get something done. But you need someone in another department to make a change. You have no authority over them.
For some reason, they won’t do it. Maybe they don’t care. Maybe they’re too busy. What’s one to do?
The solution is to really get to know the other person.
It’s very likely they feel you don’t understand their role, their department, their workload, or their emotions. Because of this, they may not even hear your words. They have other priorities. But if you get to know them over time, they will feel you understand them. And only then will they really start to hear you.
There’s no way to circumvent building relationships within culture. Better to start now. Before there’s a problem.