Don't know if it's used in BrE, but as you said, it's very common in AmE. It's used as an exclamation of despair or frustration, when you've reached the end of your rope: "For crying out loud, would you kids stop whining and just shut up?!"
"Go clean out the grease trap."
"I've cleaned it out three times this month, for crying out loud. Let Fred do it this time."
"Please hold while I transfer your call."
"Oh, for crying out loud, I've been transferred three times already! Can I please speak to someone who knows what they're doing?!"
Student or Learner