Thanks for your post. Simple questions, but unfortunately there aren't simple answers.
There are a number of surgical options for nystagmus, some cosmetic to move the null point, some to try to improve vision. Assessing whether someone's vision has improved after nystagmus surgery is difficult. It's not just a question of whether you can read further down the eye test chart. For that reason doctors don't always agree on the effectiveness of nystagmus surgery. Few doctors anywhere in the world do nystagmus surgery and there has been little research.
Yes, nystagmus surgery is available in the UK. I would suggest you contact Professor Irene Gottlob in Leicester to see whether you are suitable.