In many parts of the country you can be referred to your local hospital dermatology department for assessment and treatment but some areas of the country this is not funded by the NHS. Some treatments such as botox are not generally funded by the NHS so patients with hyperhidrosis will seek private treatment.
The Consulting Room maintains a list of clinics that provide treatment for hyperhidrosis in the UK and Ireland which is searchable. They also have a page where you can find miraDry clinics for treatment of the underarms.
The International Hyperhidrosis Society also has a searchable physician finder on their website. You can search by clicking on the image below and even filter your search for body area and/or treatments offered. Like many specialist clinics in the UK they tend to be clustered around larger towns and cities so you might have to travel.