I did the measurements, and I have to say I am a bit surprised on was I saw.
This was the setup (click on any photo below to get it big):
I configured Arduino Due to generate a 2.5KHz PWM frequency, fast enough to get many pulses, and slow enough to not run into potentially high frequency issues. The same signal was measured directly (channel 0) and with a delay by passing through the 250m of copper wire (channel 1). This is the view on successive pulses.
It shows a little delay an falling edge, followed by an additional peak. The rising edge does not show a delay (at that resolution).
So let me start with analyzing the falling edge and the peak, zoomed in here:
The delay on falling edge (2.4μs * 0.6 = 1.44μs) is something I think I can explain. The speed of passing 1m of the copper wire is 1000000/(1.44/250) = 173611111.1m/s or 173,611km/s, slightly less than 60% of speed of light.
The peak occurs 7.44μs after the delayed falling edge. I don't have an explanation, maybe the 250m have some coil effect?
Here I zoomed into the rising edge to get timings at 2.5ns resolution of 400Msps logic analyzer:
A delay of only 5ns for 250m would correspond to 167 times the speed of light, and Einstein showed that this is not possible. Here the copper cable coil effect must be the cause.
Copper cable, at least in coil form, is not good for getting signal delay and therefore phase shift. I will cut 1m from copper wire and see whether delay is linear (should give a delay like 1.44/250 = 5.76ns).
Explanations for the peak following the delayed falling edge and the super minimal delay for 250m copper wire on rising edge are appreciated.