Start here, by taking a look at the forms, to see whether you want to try to tackle the task yourself:
https://www.gov.uk/government/publications/registered-titles-part-transfer-tp1
While it's usual for two parties involved in a contract to use different solicitors (if they use solicitors at all), it's possible with simple transactions like yours for both parties to use the same solicitor, thus cutting down on fees. (When my neighbour asked if he could buy part of my garden, his ex-brother-in-law acted for
both of us in the transaction, with my neighbour agreeing to pay all his fees).
Take great care to get the alteration to the title plan absolutely accurate though. The draft document that the solicitor sent to me didn't have the new boundary line exactly where my neighbour and I had agreed that it should go, so I had to revise the document and send it back to him. Then, several years later, when my neighbour came to sell his house, I understand that there were still queries raised by the buyer's solicitor about the exact location of the boundary. (I wouldn't be at all surprised if similar problems arise when my house is sold but, as I'll probably be dead by then, it will be for my executors to worry about, and not me!)