What is the grammar behind this sentence?
One wild cart ride later they stood blinking in the sunlight outside Gringotts.
The problem is with the words preceding "they." Is "one wild cart ride later" a complete phrase? Is it an adverb like "after a wild cart ride" in the sentence:
After a wild cart ride, they stood blinking...
It is an adverbial phrase expressing a transition through time, much the same as you might say "two hours later" or "three days later".
Of course, it doesn't have the same obviousness of meaning as those, because "wild cart ride" is not a normal unit of time. It is used here creatively as if it were a unit of time, that being "the length of time it takes to finish a wild cart ride", but that's not all there is to it. It helps to express briefly something of what happened in that time. It adds colour and meaning without spending extra time on narration. Thus it means, but says more expressively and creatively, "they had a wild cart ride, and after that..."