Queen Elizabeth's mount in the photo on this thread is a Fell Pony, a breed indigenous to nothern England. It is actually a good mount for her at her age for several reasons. The Fell is a very docile breed with a good mind and would likely be a very quiet ride. Also, her leg placement is very good for her.
I have always seen her riding with a scarf on her head. I'd much prefer seeing her in a safety helmet though!