LED radiation angle appears to be changing with distance
So the angle appears to be changing with distance. I've tried several LEDs and the same happens (different measurements, but the angles still appears to change with distance).


That is normal.  You're trying to measure something like the FWHM diameter of an extended source from a short distance.  Since each point on the source is emitting light over a range of angles (which are not necessarily the same), as you measure the approximate edge of the sum of those emissions, you're going to get different values depending on how they overlap at each distance.


  The LEDs should radiate a cone of light, and this appears to be the case when I observe the light beam with a sheet of paper positioned in parallel to the LED (I see the light take the form of a triangle on the paper), so I should be able to calculate the angle with arctan().


A single mode laser would radiate a cone of light.  This is an LED, and you are much too close to approximate it as a point source, so its radiating a huge number of different cones, each spatially offset from each other.  So yes,  you can measure that angle with arctan(), but it won't be constant.  

What are you actually doing?  Is an LED the right light source for your project?