Analyzing the doppler shift to calculate speed only works if you know what the unshifted audio spectrum should be. Trains generate a ton of noise at a wide range of frequencies and that noise probably varies significantly based on a bunch of factors.

If you put the microphone directly against the track, I would bet the friction and movement of the wheels against the track generates vibration that is fairly consistent for a given speed. Maybe a sensor that better detects slight vibration would be better than a microphone for this use case.

Additionally, train engines run as generators to actually power the wheels, which means they're likely running at consistent RPMs or a consistent range of set RPMs. This could be listened for.