The Five Eyes is the most powerful spying alliance in history. Forged in the conflagration of World War II, the intelligence-sharing pact among the U.S., Canada, Britain, Australia and New Zealand helped defeat Nazism and Communism, fend off al Qaeda and Islamic State and shield its members from the ever-growing risk of cyberattacks. Now former and serving members of the Five Eyes and outside experts say it is facing its biggest challenge ever: President Trump’s pledge to curb the “Deep State” by overhauling America’s vast intelligence apparatus. Trump left the Five Eyes alone during his first term and hasn’t named...