Ghanashyam Chaudhary, a local police officer, said "hundreds of people went missing" and many may have been swept downstream into remote areas.
As many as 100 people reportedly managed to swim to safety, he added.
Pandey said helicopters had been requested to help scour the river.
The 500-m long bridge was crowded with pilgrims headed to a religious ceremony to celebrate the full moon that began on Monday, Pandey said.
"The remoteness of the area and poor communication facilities have delayed rescue efforts," he added.
The Himalayan nation is cut by dozens of rivers across one of the steepest topographies in the world.