Gresham's Law will destroy their fiat savings, since everyone will want the stronger bitcoin, devaluating fiat. We'll see it happen first in the trouble economies, with high inflation and generalized suspicion on governmental money because of past problems.
Would you mind explaining how your using Gresham's law to suggest that people will want a stronger Bitcoin?
Gresham's law states that any circulating currency consisting of both "good" and "bad" money (both forms required to be accepted at equal value under legal tender law) quickly becomes dominated by the "bad" money. This is because people spending money will hand over the "bad" coins rather than the "good" ones, keeping the "good" ones for themselves. Legal tender laws act as a form of price control. In such a case, the artificially overvalued money is preferred in exchange, because people prefer to save rather than exchange the artificially demoted one (which they actually value higher).
If I have it the right way round, fiat will be the overvalued currency ('bad') and Bitcoin will be the undervalued currency ('good'). In effect it'll mean that everyone will hoard BTC rather than spend it making it useless as a currency because nobody will bother with it as there will be no supporting infrastructure. In effect, it'll become just another commodity...
In that notion, "dominated by" means that the bad money will be the one everyone will be given away, so it will be the most transacted one. Because the good money will be hoarded. So, if you received payments in fiat and bitcoin, you will keep bitcoin and use fiat to buy things. Of course, that will make the exchange rate of fiat go down (devaluate) and the one of bitcoin to go up. We already are seeing this phenomena. It's this that is justifying the rise in price of bitcoin.