Seems we are not talking entirely about the same thing. I think you have set the GAME_UPDATE_time to a fix time, and then using a speed variable to change mobile moving speed.
But for me game speed is not just about mobile moving. For example, mage chant spells, it gets delayed. So if game speed changed, the delay time should also change. Here, you will use system time * Game.speed to measure delay time?
You mean you want game to run at 1/4 speed for example? The only way I know is to have a global variable like speed and multiply every operation you do in your game by that variable.
So if you want to have 1/4 speed, have like static float speed = 0.25f;
When you want to move character, multiply the amount you add to players position by 0.25f. So
player.x += player.speed * Game.speed;
And yes, you need to do this for EVERY variable you modify that needs to change speed.. I don't know any other way.