While the world’s attention has focused on Iran’s nuclear ambitions, other players in the Middle East have been laying their own plans to develop nuclear power to meet future energy needs.

Saudi Arabia, the most ambitious of the group, has announced plans to build 16 reactors over the next several decades, providing a projected 15% of the country’s electricity possibly as early as 2032, according to a Saudi government website. The estimated cost of the program: more than $80 billion, according to an analysis of the plans by...