YMMV quite a bit on this. When I did the apostille the first time, Monument Visa turned it around in about a month. Others have had longer delays. It just depends, it seems, though there was a period when turnaround at DeptState was clearly bad.
My lawyer submitted the application with an expired fingerprint apostille; it had just expired a couple weeks ago. She requested I get them re-done and have it ready by the time they expect SEF to actually get around to looking at the paperwork, in about 6 weeks. So there’s another YMMV there on how your lawyer works and what they’re willing to do and not do.
You probably cannot use one that’s done locally simply because SEF is probably going to want to see one from the FBI, not from a local agency.
For what it’s worth, look at using one of the “FBI channeler” services for fingerprinting. Fieldprint has a number of offices scattered around in various shipping/notary places. We had one nearby. Filled in the forms online, paid $50, drove over, nice woman fed our hands into the scanner, out the door in 10 minutes. Got results back from FBI via email in 3 hours - we literally just got back, stopped for dinner and to hit Big Lots, email with the PIN came in just as I was sitting down to catch up. Waiting for the wife’s now so I can pop the PDF off to Monument Visa tonight. USPS has fingerprinting locations now too, but I paid $50 + $18 and it took 2 days to get the results from the FBI. shrug
To Nevada’s point - I fedexed the bank paperwork. The fund paperwork is going cheaper prio mail because despite a “please send wet-signed paperwork within 15 days” admonishment, both went forward based on scanned paperwork and the bank’s instructions, the shares are officially registered, and neither has said a peep otherwise, so they’ll get the signed paperwork when it gets there. 
The lawyer said she’s fine with scanned documents and simply wants me to re-sign her POA when I show up for the biometrics, so I’m not wasting $0.01 on postage for those. There’s certain things to not dick around about and others that simply don’t matter, the trick is figuring out which is which.
for more, check the apostille thread -