CAIRO—Three years ago, President Abdel Fattah Al Sisi’s government announced that a gleaming new capital city would rise in Egypt’s eastern desert by 2022, featuring tree-lined boulevards, new homes for five million people and the tallest building in Africa.

The project is now well behind schedule, according to its military-controlled developer. The only finished structure is a military-owned hotel in a cream-colored compound. Project spokesman Khaled El Husseiny said just one of three phases is under construction. “We did...