Report the change promptly through the channel applicable to the current stage. Include the amendment with records still being prepared; if the case is already submitted, check how to provide the update. A scheduled interview does not mean waiting until that appointment to report a relevant change.
Matching the update to the current stage of the case
Delay is what creates friction: a fee receipt in the old name, a filing in the new one and civil documents in a third variation force a verification query nobody planned for. The published description of processing in weeks is a description, not a promise, and the interview step still depends on category determination and visa availability. Neither timing nor approval is guaranteed by an orderly paper trail — it simply removes one avoidable delay.
Include the legal document that evidences the name change itself, not just a statement that a change occurred, along with a short cover note cross-referencing the case identifiers used under both the old and new names. Notify every party that issued a separate document under the earlier name, not only the main filing, since a record corrected in one place while another still shows the earlier name recreates the same mismatch the update was meant to resolve. A single consistent update across all affected records does more to prevent a verification query than speed ever could.