If you own property in the UK, HM Land Registry holds a record of your ownership. After a deed poll, you should update the registered name so that it matches your current legal name. The process does not change who owns the property; it simply updates the name on the official record. You will need to complete a Land Registry form and submit it along with a certified copy of your deed poll (and, if applicable, any marriage certificate, decree absolute, or other supporting document).
The relevant form is AP1 (Application to Change the Register) for registered land in England and Wales. There is no Land Registry fee specifically for a name change (as opposed to a property transfer), though standard registration fees apply if other changes are being made at the same time. You can submit the form by post to HM Land Registry or, in some cases, through a solicitor or conveyancer. Many people choose to update the Land Registry when they next remortgage, as the solicitor or conveyancer handles the paperwork at that point.
Your mortgage lender should also be notified separately. The mortgage account is a contract between you and the lender, and the name on that contract should reflect your legal name. Contact your lender directly with a certified copy of your deed poll to request the update. This is usually handled by their customer services or mortgage administration team. The mortgage terms, interest rate, and payments are completely unaffected.