It is hard to say how long it may take to receive the document requested: usually it's about 6 to 8 weeks or more (*). Italian public sources will be solicited after 5 weeks, first by fax or email and then by telephone. Ultimately, you will either receive the record(s) requested or a notification of No Record Found.
(*) Please note: With the new Law No. 113/2018 (which modifies the legislative framework set by the Law No. 91/1992), the processing time for Italian Town Halls to issue civil vital records (births, marriages, etc.) requested for the purpose of filing for Italian Dual Citizenship is now up to SIX (6) months.