Let's start with "sorry"
By the time the phone rings, there's already trouble. When that manager is called or this department is reached, it's because someone is disappointed, angry or stuck. Illness, broken promises or a real urgency have led to this new conversation even taking place.
So don't start with, "[Name of company] mumble mumble" as if there's a blank slate just waiting to be written on. There's already a lot of writing on that slate. Don't demand to know the record number or begin with doubt and an edge of dismissal. Be on our team.
"It sounds like we've got a situation on our hands..." is a fine way to disarm the person you're about to talk with. He won't have to spend the first six sentences expressing his anger and urgency, because in less than ten words, you've done it for him. Or perhaps, "I'd like to help, if you'll bring me up to speed..."
It's not easy being on the receiving end of a days'-long parade of blame, but no one said it was easy. We asked you to do it because you're good and because it's important, not because it's fun.