Interesting question, and I haven't rtft to see if it's been answered yet, but from first-hand knowledge of horses on snow... they cannot easily cope in snow if shod in the usual way. Snow kinds of packs into their hooves (in the soles of the feet, if you like[, and it tends to build up a kind of wobbly platform that they have trouble walking on (let alone galloping and executing sharp turns[.
My horse is unshod though, so does not have this issue. And of course he is not used for polo!
I imagine if you're playing polo in the snow, you'd need your horse to either be unshod (which is a whole long project to get right, if you wish the horse to remain comfortable at all times[, or you'd need, I suppose, to fit special shoes.
I have said in past threads how much I dislike polo, just as pps have stated and for the same reasons. That applies to whoever is playing - royal or not!