Hyperinflation.
Everyone will want to get rid of their remaining fiat currency as quickly as possible. Many people will end up with bags of dollar bills and nothing to eat.
Maybe the governments will step in and offer to buy fiat currency at a fixed rate (and subsequently destroy the bills and coins) using tax Bitcoins, so fiat currency holders won't end up starving.
The transition period will feel like after a world war. After a few years the wounds will have healed, power and welth will have shifted to new owners, just like a few years after the world wars.

Philipp