Snipping off just the first few kilometers is not catastrophically destructive yet, and cutting it down further up would require multistage rocket designs, sophisticated steering/targeting and potentially significant yield (you'd need to cut unobtainium, after all...). If you can build a space elevator, you can defend against those.
You better thoroughly inspect what cargo you put on the elevator itself, of course.
Not to mention, securing the cargo would be an extremely difficult task in itself, especially when one of the main thinga you'd like to raise through the space elevators is rocket fuel.
and what, 12" wide? 24"? that's still very difficult to target
But yes, a space elevator would be difficult to defend in World War III.