I agree. There's some passive aggressiveness that really isn't necessary. 100% you have to be direct with folks, and knock off all the placating, but I'd have worded it like this:
"Co-worker", I need you to consistently respond to emails regarding your cases. When people are requesting information, timely communication is part of the job and is necessary for the team to function effectively. This has been an ongoing issue and I need to see immediate improvement going forward.
If this continues to be a problem, you leave me no choice, but to escalate the matter. (this shows it's the employees actions that will result in the escalation)