how long does it take a branch of The Northfork Bank in New York  to
cash an out-of-state check for a business account holder?

Normally, banks put a 10 day hold on out-of-state checks.

If you want the check cashed or deposited immediately, when you take
it to the bank, don't go to a teller. Take it to an account
representative (yours if you have one) and ask them to sign off on it,
that you want the money to hit your account as if it was a local
check, and that you want the funds, today. They will call the bank
it's drawn on and initial the check. Then you can go through the
teller, and the normal 10 day hold will be off.