"The UN" is less a thing than a differentiated body of representative assemblies that usually have to vote on action. China has whipped considerable support even within the Human Rights Council, and wields a veto on the Security Council. Even getting the International Court of Justice to issue an advisory opinion would probably require winning a General Assembly vote among states China has often bought off.
The most the UN Secretariat could do by itself is appoint a Special Envoy to oversee human rights in HK. But that would probably anger China and impede future cooperation. Plus the envoy wouldn't have much power beyond issuing reports.
The "registered with the UN" line is usually trotted out because China denies the legitimacy of the Sino-British Joint Declaration as a treaty, so that helps establish its legitimacy.