LIMA, Peru—Oscar Villarroel confidently told his customers at a barbershop here in recent months that the Venezuelan regime would soon fall and he could be heading home.

Now, with autocratic President Nicolás Maduroresisting mounting pressure to step down, Mr. Villarroel and others in the vast Venezuelan diaspora of 3.4 million are wondering when, if ever, they will return. They are bracing for many more exiles to join their ranks.

“We’re...