As soon as he is sworn in.
When the Secretary of State issues his commission. Hopefully that would be within an hour or two of the result of the confirmation vote. However, it doesn’t have to be, and is up to the Secretary of State to actually prepare and sign a certificate that entitles Kavanaugh to take his seat on the Supreme Court.
This was the very issue that spawned the famous Supreme Court decision of two centuries ago, Marbury v. Madison. The Secretary of State refused to issue a commission for a magistrate.