I need some proof that the electric (E-field) and the magnetic component (H-field) of a lightning stroke both propagate at the speed of light, and that there is not a significant delay between them.
The E and H fields of all RF signals travel at exactly the same speed - the speed of light. The phase relationship between the two fields remains constant as they travel through various mediums.
Lightning in itself is not an RF signal but it does emit RF (with E&H waves) as it propagates.
After reading some of the additional dialog provided by the OP in response to my answer, I believe the other party is talking about detecting the E field of a capacitor with the storm system being one "plate" and the earth forming the other "plate" of the capacitor. Note that this is quite different from the paired oscillating E and H fields of an RF signal propagating through space.
Inside of an ideal, charged capacitor there exists only an E field that is measured in Coulombs (shown as a unit of C). One proton has a charge of 1.6 x 10-19 C.
For a capacitor where the diameter of the plates are substantially larger than the gap between the plates, there will exist an equal charge in the space between the plates. When this charge reaches 3x106 N/C, the air inside the capacitor will break down and a spark (lightning in this case) is formed.
So it seems that the proposal is to measure the electric field of the capacitor formed by the storm system and the earth as a predictor of the potential of lightning. The referenced probe would form another plate in the capacitor - not to be confused with an antenna. This may work in principle although there will be a substantial difference in the E field as measured when between the plates (inside the storm system) and outside of the plates (near the edge of the storm). If prediction is not the goal, then edge detection of the collapsing E field, as a result of a dielectric breakdown of the air exhibited as lightning, may be a solution.