The United Nations Digital Library provides access to individual speeches contained within meeting records of the Principal organs. Speeches since the mid-1980s or earlier, depending on the body, can be searched by country or organization, speaker, document symbol, or topic. Historic data is added regularly.
The Index to Proceedings (ITP) is a powerful tool which provides a window into the annual meetings of the main UN organs. The publication is comprised of a subject index to all the documents issued by the body in question during a particular session/year and an index to speeches delivered before the forum in question during a particular session/year.
Ask DAG has entries for each Member State's statements in the General Assembly's general debate from 1946 to today. Search by country name.