Uhh, I thought Brooklyn was pretty fast, too. I got my approval on Jan 24, so I guess the oath ain't happening fast. If you don't mind, update here when you get scheduled.
Covid vaccine was available when I had an exam so I didn't need a blanket waiver for covid. I had two blanket waivers, one for a vaccine that was not available. (similar case to yours) and one for the vaccine that required three doses to be fully immune but they only gave me one dose and noted that there was not enough time to administer the second and third dose. They are pretty accommodating so I would assume you won't face any issues.
If vaccine is truly unavailable they will issue a blanket waiver and won’t put a blame on you. That was my experience with medical examination in Tbilisi.