Does the statement, "By Allah, regarding my children, I will not live with you," which is intended as a divorce, take effect merely by retracting it via a message, or does it only take effect upon the actual relocation to the shared dwelling?

This wording is not an explicit divorce. If he did not intend divorce by it, but rather intended to prevent himself from living with his brother, then if he lives with him, an expiation for a broken oath (kaffarat yameen) becomes due, and no divorce occurs. However, if he intended divorce, then divorce occurs if he lives with him.