Is there a way to get the timing when EditButtonitem in UITableView is pressed by a delivery method?

UITableView does not have the editButtonItem property, so we assume you are talking about the UITableViewController, but in this case, from UITableViewController Class Reference

It implements the superclass method setEditing: animated: so if a user tapes an Edit | Done button in the navigation bar, the controller toggle the edit mode of the table.

setEditing(_:animated:) can be overridden.