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.
Only if the material is way over provisioned. In general, the higher the intrinsic structural load is on a material, the easier it will be to destroy. So, to defend from these attacks, you not only need a cable that can support its own weight, plus the weight of the desired payload, plus some small-is extra tolerance. Instead, you probably need a cable that can support, say, twice its own weight plus four-five times the payload. Not to mention, now you don't only need excellent strength along the cable, but also across from it, and extreme heat resistance too (all of the strength is irrelevant if it's enough to coat some part of the cable in thermite and ignite it)
You only need to defend the easy to reach parts. So the base and the cargo pod. To hit the upper parts you need advanced rockets and targeting systems.
Why advanced? It's a stationary target that's 35,000 km long. I don't think it would be that hard to hit.
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.
In general, the more tensile strength you want in a cable made of a given material, the thicker you need to make that cable. Now sure, we can imagine whatever magical properties we want of our space elevator cable material, since no known material that could do this exists anyway. But it's far more likely that you'd need a cable that's a kilometer or more in diameter to achieve the tensile strength needed to support its weight at 35000 km of length, than it is to be a few inches wide.
Yeah, but then we're discussing opponents with nuclear arsenals and ICBM programs. Those opponents are generally reluctant to nuke stuff, or commit acts of war similar to nuking stuff.
But yes, a space elevator would be difficult to defend in World War III.
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.