If we're already in September, more than 30 days have passed since May, so your employer has already well exceeded the legal deadline - they can't use "I need to wait" as an excuse. What you're saying is right: by law they have 30 days from the end of employment, period. As for the work certification (the CU form or certificate), that's a different thing and they have to give it to you anyway regardless of the TFR, it's a document you need for INPS.
If you haven't received anything, send a PEC to your employer asking for both the TFR and the certification, with a copy to a union or a labor consulting firm. Often just an official letter is enough to get them moving.