As U.S. and Iranian diplomats inched toward progress on Tehran’s nuclear program last week, Saudi Arabia quietly signed its own nuclear-cooperation agreement with South Korea. That agreement, along with recent comments from Saudi officials and royals, is raising concerns on Capitol Hill and among U.S. allies that a deal with Iran, rather than stanching the spread of nuclear technologies, risks fueling it. Saudi Arabia’s former intelligence chief, Prince Turki al-Faisal, a member of the royal family, has publicly warned in recent months that Riyadh will seek to match the nuclear capabilities Iran is allowed to maintain as part of any...