Your council tax account is managed by your local council and should reflect your current legal name. After a deed poll, contact your council's council tax department by phone, online, or in writing, and provide a certified copy of your deed poll. Most councils will update your records quickly; some may ask you to fill in a short change of details form. Your council tax bill, direct debit, and correspondence will then be issued in your new name.
If you live in a property with other adults, only your own name needs to be updated on your own records. If you are the named account holder responsible for the council tax bill, the update ensures the bill is in your correct name. If the property has a joint liability (both names on the bill), let the council know which name has changed and provide the deed poll as evidence.
For anyone moving house at the same time as changing their name, this is the ideal moment to set up a new council tax account in your new name. You will need to register with your new council anyway; simply use your new name and, if asked, provide your deed poll. Some councils share data with the electoral roll and other local authority databases; ensuring your name is correct on council tax records helps keep all local authority records consistent.