My cursory internet research into 19th century Polo is turning up vague results, variously stating that Polo was first played in England by the 10th Hussars in 1834 and also, not played in England until after the Crimean war in the 1860s. Did soldiers in Crimea play polo in their down time, or would they not even have heard of the game?
Polo was developed or invented as a game or sport in Persia.
The British encountered it in India.
The Crimean War was from 1853-1856.
The first European polo team was created in India by British Lieutenant Joseph Ford Sherer in 1859. The first British Polo club was established by Sherer in India in 1862. The first set of formal British rules for polo were established by the Hurlingham Club in 1874.
So, polo as an organized British sport did not exist until after the Crimean War. It was unlikely, therefor, that it was played by British Cavalry during that war.
Unlikely, but not impossible. As you note, a form of polo is claimed to have been played by the 10th Hussars in Aldershot in 1834. However, polo did not really get organized as a British sport until several decades later.