Short answer: No, US segregation law did not apply in the UN Headquarters. So black/brown leaders, ambassadors and represantatives were allowed in the whole UN HQ and its other buildings.

Long answer:

United Nations is extraterritorial, meaning it is an international territory and people are exempt from the legal jurisdiction of the country in which they reside.

The site of UN Headquarters in New York/USA is owned by the United Nations. (Think it as any other foreign country embassy).

No federal, state or local officer or official of the United States of America, whether administrative, judicial, military or police, may enter UN Headquarters, except with the consent of and under conditions agreed to by the Secretary-General of the Organization.

You can read the history of the UN Headquarters building and some other interesting stuff from this official document titled FACT SHEET: HISTORY of UNITED NATIONS HEADQUARTERS.

Related content from the linked document above:

However, the United Nations is bound by an agreement with the United States, its host country, to prevent its Headquarters from being used as a refuge for persons attempting to avoid arrest under the Federal, State or local laws of the United States. People being extradited by the United States Government are also denied use of United Nations Headquarters in attempts to avoid arrest. The UN has its own fire-fighting and security forces, and its own post office branch.