It's the same question as when one opponent is ready to nuke enemy cities while the other is not. In theory everything turns out the same way you postulate. In practice cost of winning the war in this manner might be higher then the cost of losing localized, limited conflict.

That's why there's never been a war between two nuclear-armed countries: everybody is shitting in their pants because of exactly that logic. Presumably we'll find out when the PRC invades Taiwan.

However, if that doesn't end up with a radioactive wasteland, we'll probably find out shortly afterwards what happens when every two-bit drug cartel or extortion mafia can cheaply assassinate anyone, anywhere in the world, with an ambush drone.