In the days before fridges, large houses would have an ice well or house. These were pits with a circular roof to allow ventilation. They were often brick lined and packed with straw. The ice would last for month. See more on the London ice houses.
The Hampton ice house is situated in Home Park by Hampton Court. It is situated to the left of the path from the Hampton Wick entrance.
It was built 1625-26, commissioned by King James. It is 12 sided and sits on a small mound. This Country Life article says "It is known to have been a brick-lined round well, 30 ft (9.1 m) deep and 16 ft (4.8m) wide and covered with a thatched timber building. "
I can't find any reports that give more info about this ice house. As it is located near a pond, ice could have been taken from there. Alternatively ice could have come by boat on the Thames, although I think imported ice from Norway and America only started in the 1800s). These are just my ideas.