It really varies, out here a few dozen miles outside NYC, best I get is 2 day delivery. I used to have next day pickup a few years ago when Amazon had their own lockerbox nearby.
However, last time I ordered something to a relative in NYC, I was surprised to find that they even have same day delivery for many items.
I'm in London, and next day delivery is the norm, with same day delivery increasingly common. Two day delivery is in the "why am I even bothering with Amazon" territory.
Our Amazon delivery people regularly drop off 4-5 orders with neighbours before they get back in their car...
For me 2 day delivery is still pretty good since the alternative tends to be either waiting a week or two, or putting in a lot more time and effort to go get the thing in person (most likely by taking the entire day to go to and come back from NYC).
However, last time I ordered something to a relative in NYC, I was surprised to find that they even have same day delivery for many items.