Christmas Common is composed of the Lewes Nodular Chalk Formation, Seaford Chalk Formation And Newhaven Chalk Formation (~71 to 94 million years old; Cretaceous period).
The chalk is described here - http://en.wikipedia.org/wiki/Chalk_Group
You'll be able to guess from those names that the same strata crop out on the south coast. A couple of papers describing the rocks and mentioning some of their fossils
This is an excellent resource on echinoids
Where I grew up, on the North Downs in Surrey, in the copses within or at the edges of the chalky fields, there were piles of flints resulting from the farmers raking the fields. Some of those flints were echinoids. It is much more resistant to weathering than the chalk. Your specimen will also be made of flint.