Colonel
Lieutenant-Colonel
Major
Captain
Lieutenant
Cornet
Regimental Serjeant-Major (After 1813)
(Troop) Serjeant-Major (Troop after 1813)
Serjeant
Corporal
Trooper
That would be the proper structure for a british cavalry regiment that is not a part of the household cavalry. Before 1813 there would only be one Serjeant-Major per regiment, so accordingly there was no need to differentiate between troop or regimental serjeant-majors.
For the household cavalry it would be the following:
Colonel
Lieutenant-Colonel
Major
Captain
Lieutenant
Cornet
Regimental Corporal-Major (after 1813)
Corporal-Major
Corporal of Horse
Lance Corporal of Horse(/Corporal?)
Trooper
I am not that sure about these as I am of the normal cavalry ones, since I only found the modern ranks, and extrapolated the stuff about the other cavalry ranks onto these.
Also, will you update the explanations regarding CdB and Major for the french ranks?