I think next-door neighbour is pretty widespread. And they may still be the next door to yours, even if there is grass and fence in between!

I wonder if the term emanates from the terraced streets of industrial England, where the person who lived in the next house was, literally, at the next door, and where doors were in pairs all the way down the street?
In the early days, houses weren't numbered, so anyone looking for a person would have to ask where they lived. If they got hings* slightly wrong, they would be told, "Try the next door."

* EDIT - that typo should be recast as "Things", rather than "hingEs"