The met people apparently have lasers that they fire into the air, then they use trigonometry to find the altitude of the bright point where the laser hits the cloud.
You could use a fireworks rocket and count the seconds until it disappears... But not at the airfield!
Perhaps the more experienced people have some rules of thumb. I can imagine that if you know the windspeed and there are some cloud features, you could use that to guide your guesstimates.
Sounds like a scary ride - glad you got away with it.