WASHINGTON—As time ran out Thursday for Congress to block the Iran nuclear deal, the Obama administration began moving assertively to put it into effect, appointing an official to oversee its implementation and starting the process of lifting punitive economic sanctions against Tehran.

Congress had a 60-day period in which to vote to disapprove of the nuclear deal, reached between Iran, the U.S. and five other world powers in July. As that clock expired, U.S. officials named Oct. 18 as “adoption day” for the nuclear accord,...