Find nearest neighbour in a radius
The K-D tree is a good data structure for solving this. However you can't blindly apply the search procedure only to the center point, you must be a bit smarter.

While searching the K-D tree for your points, every time you must choose the left or right child to search in, check whether the splitting plane is to the left of your circle, intersects it, or is to the right of the circle.

If the splitting plane is to the left of your circle you must continue searching in the right child, and vice versa. However when the splitting plane intersects your circle you must search both children.