No, it does not. Fat freezing is perfectly safe; there are no known instances of it causing nerve damage.
Your fear probably comes from the idea that cooling down the fat is somehow aggressive against your body. And it is true that the fat freezing process will cool your tissue down to sub-zero temperatures – it usually operates with temperatures around -6ºC. That is indeed very cold, but not unnaturally cold. For a bit of perspective, this is right about the temperature range of the water in lakes where people go ice swimming, which is a tradition in Russia and other eastern European regions.
Other potential side-effects include redness, bruising, minor swelling, skin sensitivity, and intermittent numbness in the area that was treated. None of those are fun, but they are minor if compared to other methods of dealing with stubborn fat, such as liposuctions.
So you will be fine, don’t worry. Now that you are more relaxed about this whole thing, why don’t you click this link? It will direct you to professionals who do fat freezing in Kent.