If you suspect a serious injury, just stop the game. You may also consider the location of the injury. If you have an injured player lying motionless on the ground in front of the goal, you have even more reasons to stop the game right away.
In the video below, you will see two players from the same team ending up on the ground for suspected head injuries. They are in front of the goal inside the penalty area. If the attack continues, it will be very dangerous. These injured players are on the ground and they could easily be stepped on, etc. causing additional injuries. If the ball possession changes and the defending team becomes the attacking team, you are allowing them to take advantage of suspected serious injuries in order to create a numerical advantage.
This means beyond the fact that we need to attend to serious injuries, there is nothing to be gained by the referee crew by allowing the play to continue.