You can try adding a Google Apps Script to capture when a cell is edited and add a timestamp to a different cell. Here's a previous answer that is similar: Google Spreadsheet Timestamp?

function onEdit() {
 var s = SpreadsheetApp.getActiveSheet();
 if( s.getName() == "Sheet1" ) { //checks that we're on the correct sheet
   var r = s.getActiveCell();
   if( r.getColumn() == 13 ) { //checks the column
     var nextCell = r.offset(0, 1);
     if( nextCell.getValue() === '' ) //is empty?
       var time = new Date();
       time = Utilities.formatDate(time, "GMT", "HH:mm:ss");
       nextCell.setValue(time);
   };
 };
}

You can try adding a Google Apps Script to capture when a cell is edited and add a timestamp to a different cell. Here's a previous answer that is similar: Google Spreadsheet Timestamp?

function onEdit() {
 var s = SpreadsheetApp.getActiveSheet();
 if( s.getName() == "Sheet1" ) { //checks that we're on the correct sheet
   var r = s.getActiveCell();
   if( r.getColumn() == 13 ) { //checks the column
     var nextCell = r.offset(0, 1);
     if( nextCell.getValue() === '' ) //is empty?
       var time = new Date();
       time = Utilities.formatDate(time, "GMT", "HH:mm:ss");
       nextCell.setValue(time);
   };
 };
}