ELSES grades the exams and recommends a cut score. This information is sent to the states, which rubber stamp their approval based on ELSES. The state is who actually issues the licenses.
I did call my state's board a week or so ago. She told me that at least in Vermont, examinees got a letter from ELSES before the state even got any info. She said this was generally in early June, and that she issued licenses last year around June 10.
So, I'm still thinking first or second week of June.