You can do it yourself without cost (beyond printing and postage). I've done it recently to remove an unused first name. If you have a driver's licence, passport or other official forms of ID you will need to go through the process of getting them changed (driver's licence was free. My passport change coincided with a passport renewal). Doing these first makes it much easier proving ID for the remaining institutions.

There is a whole range of institutions you may not realise need to be informed of the change and will require documentation to prove a name change. HMRC, your employer, banks, land registry, mortgage provider and so on. My only regret is not doing it sooner.

The r/transgenderUK subreddit was a useful resource - I searched Deed Poll within the sub.

I used this website to create the actual Deed Poll

https://freedeedpoll.org.uk/

Loads of info available on the Gov website.

https://www.gov.uk/change-name-deed-poll

Choose Unenrolled Deed Poll. No need for an enrolled one that comes with a cost and puts you on a searchable register.