I presume that your drawer's widgets will have buttons or listtiles for navigation. So, you just need to add a timer.cancel on the "onTap" function

Drawer
   ListView
      ListItem
         onTap() {
            timer.cancel(); // <-- this will cancel your timer
            Navigator.push();
         }
      ListItem
         onTap() {
            timer.cancel(); 
            Navigator.push();
         }

Cheers