It usually only takes a couple of days to get an appointment, maybe a week at most - just depends what they have available.
You can choose which Application Support Center to go to, so if there aren't any good appointments at the NYC center, and there's another center you can get to instead, you can always book an appointment at the other one.
Some people have also had luck booking an appointment for a few days ahead, but then just turning up at the centre a couple of days earlier as a walk-in.
The appointment is very straightforward and shouldn't take more than 5-10 minutes. It's done by USCIS on behalf of UKVI, so they are just there to take your fingerprints and photo and nothing else - they don't care what visa you're applying for or what documents you are sending.
So, you just take with you your passport and the printed biometrics appointment confirmation. When you get there, they will check your ID, make sure it matches the appointment confirmation, and will then stamp the confirmation sheet. Then they will take your fingerprints and a digital photo, and send you home.
Once you have finished the appointment, you just place your passport and stamped biometrics confirmation in the package with your other documents and mail it to Sheffield within 5 days.