This thread's not really about fuses...
And there seem to be some misconceptions, too.
Note that if you place a '12volt LED' (LED+current limiting resistor in series) across a fuse holder, it will light if the load side shorts to ground and the fuse blows.
You should measure little to no voltage on the load side of the fuse/LED, since that side is now shorted to ground. A partial short to ground (bad enough to blow the fuse, but not a dead short) *might* light the LED and still allow measurable voltage on the load side of the fuse/LED pair.
Assuming a single load on the fuse: If you then remove the load side short and the load itself (as in, disconnect the bad component that shorted to ground), the LED will no longer light. There is no longer a complete current path back to the power source. In this case, you *will* measure very near the supply voltage on both sides of the LED.
If, during troubleshooting, the fault is cleared by removing the defective component, you may get measurements you don't expect, if you're not well acquainted with electrical properties.