You can do it yourself. It’s is a bit tricky but not that hard. You need to fill out the right form, TP1 iirc, then file it with the land registry. You’ll also need a reasonable drawing of the changes on the correct map (which you can download from the land registry). It is much easier to do in person if there is an office nearby. You both need to go (buyer and seller) and take ID, pay the transfer fee. It was about £40 last time I did it and took about an hour.
Or a solicitor can do it all. Probably not cheap but maybe worth getting a quote.