So now (I'm assuming) they've put out the application for the Certificate of Sponsorship. They said "should go through quickly" but I think American and British concepts of "quick" are very different....
It just depends how efficient the company is at applying for it and how quickly the certificate can be processed by UKVI. I would probably allow up to about a month for the certificate to come through, although it could be faster.
Now looking ahead - how quick between applying online and paying do you get a Biometrics appt and how quick from there can you mail in the app?
It's all up to you how long it takes - because you decide which biometrics center to attend and which date you book the biometrics. If you gather up all your documents and just wait for the Certificate of Sponsorship before applying online, you can probably get it all done within a few days.
There are 129 possible biometrics centers that you can choose from, so you just need to find the most convenient one for you with the date you need. Most people manage to get a biometrics appointment for within a couple of days of applying online.
Then, if you have everything ready to mail to NYC, you can always attend the biometrics appointment and mail the application on your way home from the appointment.
If you want to pay for priority processing ($170), you can purchase it before you mail the application, and it will get the package put to the front of the line for faster processing.