I cannot offer you any information on this matter but I think that, given the perennial shortage of horses during the period, that the RHA Troop that became the Chestnut Troop probably used what they could get.

Since the Chestnut Troop is tasked with ceremonial duties in London, that is firing salutes on days of celebration, I would suspect that the story that Wellington gave it its name is entirely mythic. In the late 19th century British army, when the name probably arose, any method of linking a newly-minted tradition with Wellington, true or not, was put into circulation.

This reminds me very much of the myth that the British Guards were given the privilege of wearing bearskins because they defeated the Imperial Guard at Waterloo. Anyone who knows anything about uniforms knows that to be inaccurate but the myth is still encountered today.

DE Graves