If just one country has it, it will be safe from nuclear threats until the others catch up. However, it might still be vulnerable to conventional invasions. Because perfect ICBM interception doesn't change the equations for countries that are not nuclear and don't have a reliable ally that is.
If all powers have it and ICBMs become useless, WW3 is still possible, even more likely. After all, the Cuba crisis happened right after MAD became plausible and set a precedent that is hopefully followed in the future.
Finally, MAD is just as possible with bombers and cruise missiles, possibly launched by submarines. It's just a lot more expensive and probably less reliable. But if just, say, one or two nukes can reach their intended target, it might be enough to discourage large-scale and/or nuclear wars.
