It really depends on the condition, the AME, and whether the applicant has already done their homework (which feeds back to the AME being on top of things). Routine initial AASI cases like apnea and non-insulin diabetes can be turned around fairly quickly, in fact a good AME can sometimes gain authorization over the phone during an office visit if the applicant has all of the required info.
The big exception is anything cardiac, neurological, and (I think) mental. The FAA uses consultants to determine those cases, and the consultants only meet a few times a month. Cardiac cases tend to stretch out to 2-3 months (once all required info is in), and initial certifications on neurological cases are long ordeals in many more ways than one. But yes, when the system is working well, I've seen well-prepared SI applications for things like apnea and type II diabetes go through to issuance in about a month.
I will reiterate that one of the best things you can do in a case like this is to call the region. Apnea is a very simple SI that is almost always successful with proof of treatment, so they may be amenable to helping you along.