How far away from a wing's surface does the air stream remain unaffected? I would like to know how far above and below a wing's surface an observer must look in order to see no change to the airflow. Not simply visually but mechanically, assuming a perfect scenario. I am not particularly adept at mathematics, so if there is a rule of thumb, or a general upper boundary, that would suffice for the figure I need.

For a visual representation, a wind tunnel with smoke trails. At some point distant from the top and bottom of the wing in the tunnel, all flow lines will be identical again, implying that wings have an area of effect. How far is this point from each respective surface?

Another way to ask would be to say how far apart must biplane wings be (in a perfect scenario) for them to have no effect on each other?
A precise answer would be: Only outside of the supersonic shock cone. In subsonic flow the change has unlimited range, but dies down with the inverse of the distance.

A mathematical description is possible with the Biot-Savart law. It was first formulated for calculating the magnetic field induced by a current, but can equally well be applied to aerodynamics.

Now it depends on how much error you still accept. If you need the interference-free distance between two biplane wings, express that distance in multiples of the wingspan - that is how it scales. A few wingspans of vertical separation should be enough to make the interference between wings negligible. In flow direction the distance must be much larger: Munk's displacement law postulates that in inviscid aerodynamics the distance in flow direction between two wings makes no difference. In reality, air traffic control requires a separation of up to 8 nautical miles between heavy airplanes for the turbulence to die down or be washed away by wind.