It's the Town that delivers the hit, not the County. The new assessment amount comes from the Town and then all the entities (County, police, school, etc) base their percentages on that figure.
The timeline depends in part on when the tax roll closes. In Suffolk it's usually March 1st for changes that will affect the bill that's mailed in December. So if the new assessment figure hits the database on February 20th, say, you will see the result in your December bill; but if it doesn't hit until the summer, you probably won't see it until next year.
Again this is for Suffolk. Nassau may be different, and no doubt weird in several tax ways that I am blissfully ignorant of.