I don't see any reason for fiat to stick around, unless it's by force. Let's imagine bitcoin is accepted everywhere, what possible reason would you have to accept anything else? Perhaps a different coin is more anonymous or has some technical feature. But bitcoin could probably copy those features.
People may be 'forced' to use bitcoin eventually. Fiat has its uses and likely will for a very long time, but it could lose a lot of it's value if mass inflation happens and people could abandon it for other currencies and that's where bitcoin could flourish, though paper money or some other physical currency could still be used for those in the sticks or backwards or something.