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:
https://webapps.stackexchange.com/questions/31894/google-spreadsheet-timestamp/31942#31942

    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);
       };
     };
    }