Do I have to physically cut into my Necron Overlord's Staff of Light to upgrade it to Warscythe?
Well, it depends.  

40K by the rules is WYSIWYG(What you see is what you get).  This means that if you give a guy an ax, he needs to be modeled with something vaguely ax shaped.  If you give him a bolt pistol he needs to be holding something bolt pistol like(a holster at his hip works too)  This holds true for the war scythe as well.   

There's some wiggle room on this because the game doesn't always specifically say what advanced alien melee weapons are supposed to look like.  So if you can squint your eyes and the thing the Overload is holding is two handed and sorta scythe shaped, your're probably good.

In most friendly games, telling your opponent what the character is armed with before the start of the game is generally sufficient.  Often people like to try out different combinations of wargear, and a certain amount of proxying is tolerated.   

This is all dependent on the group of guys you play with.  Some groups may be stricter about the application of WYSIWYG than others.  Check with your gaming group to be sure.

If you plan on playing in tournaments--especially larger ones, you will probably have to go with a more explicit conversion.  Tournaments are more likely to enforce strict WYSIWYG for the sake of side-stepping potential rules arguments.