The ward register decides your parking permit, your waste bags, which street committee you are in and which letters the desk can write, so it is worth having right.
Common changes: moving into the ward, moving out, a new household at an existing address, a household splitting into two, a death, a name or date recorded wrongly years ago, and a landlord and tenant disagreeing about who is registered.
This is an appointment job. Book a slot between three and five, bring the identity documents of the people being added or changed, and bring a tenancy paper or a utility bill for the address. Twenty minutes, and the change is effective the same day.
The last case in that list is the difficult one and the ward's position is worth knowing before you come. The register records who is living at an address. It is not a statement about a tenancy, a rent, a right to be in the property or any agreement between a landlord and a tenant, and the ward will not write one that implies it. Where the two parties disagree, the desk records what it can verify and says so.
There is no charge for a correction, including one the ward itself made twenty years ago, and there are a few of those.