AMESBURY, England—Britain urged Russia to provide fresh details about the poisoning of a former double agent and his daughter in March after two U.K. nationals collapsed after exposure to the same Soviet-era nerve agent over the weekend.

The U.K. called on the Kremlin to fill in further clues about the March attack—which Britain has blamed on Russia but Moscow has denied—to help U.K. authorities keep members of the public safe.

“We...