Consider an Earth-like world where every nation in the world agreed to dismantle and divide all nuclear arsenals so that every nation was allowed only a single 50 megaton thermonuclear bomb (equipped with an ICBM delivery system), and all production facilities destroyed except a single very small facility at a "World Alliance Headquarters" in order to replace broken weapons or produce for new nations in case new countries are developed. If someone uses their nuke, it will be replaced only after a period of 10 years.
The members of the agreement decide on a very thorough annual, international investigation aimed at discovering any illegal nuclear facility, manned by investigators from as many countries as wish to participate. These investigations are regulated so that the governments are allowed to hide any classified small documents and even large weapon or vehicle prototypes from sight, just not conceal any areas large enough to house components of nuclear production facilities.
Also: Someone mentioned the systems would be useless due to interceptor missiles - To counter that, let's assume that as part of this agreement, all long range missiles (greater than a few miles) except for the single allowed nuke are banned. Otherwise the countries could just amass tens of thousands of non-nuclear missiles and have the same effect as a nuclear arsenal.
What would be the key fundamental differences between real-world nuclear warfare and this scenario?
About the broadness of the question: I'll specify that I'm asking about only the key, fundamental differences between a real-world nuclear war, not "how it would play out" in general.
Edit for more situational detail, if desired:
While the nations may be able to get away with covertly working to develop a larger nuclear arsenal, punishment for breaking the agreement includes a 10 year denuclearized period (not even the 1 nuke) and a set of harsh, pre-determined, globally enforced sanctions for the same period of time. Further attempts to re-nuclearize outside of the agreement are considered an act of war against this "world alliance" and some form of removal of the leadership with replacement based on the existing government's traditional policies as if the leadership were dead. Let's assume, as a premise for the question, that this is enough to dissuade nations from breaking the agreement.