"Co-worker, I mean this in the nicest possible way. You have to respond to emails, period end of story. It is completely unprofessional to ignore people who are asking for information on one of your cases. This was a chronic issue before and I’m done attempting to fix things with you. If this isn’t corrected, and I mean immediately, I will keep going up and up the chain of command until it is fixed."
This comes off as weak and immature, not tough and authoritative.
"Co-worker, you must respond to case stakeholders' email requests for information in a timely fashion. You have failed to do so with regard to Case 1, Stakeholder X, Case 2, Stakeholder Y, Case 2 Stakeholder Z. Notify me when this has been remedied on all accounts."
You told them the expectation, you were specific in their failures (thereby documenting where the issues were, so that can be used to in job performance reviews - or in termination proceedings), and now you have put further communication onus on them.
No threats, no whining. You don't say you're going to tell dad if they don't listen, you just tell dad if they don't listen. They know the rules and procedures ... or should, anyway.