The coins of a fiat currency, if designed correctly, will always have an intrinsic value lower than their face value; that is to say, that the value that the nation all agrees the coin is worth should be more than the cost of the materials and labour used in minting the coin in the first place. There is a very simple reason for that; if you put $2 worth of silver into a silver dollar for instance, then people just start melting down the coins and selling the silver. That's a bad way to manage currency as it means that you never really know how much currency you have out there and you end up devolving into a de facto barter system using precious metals instead of coins.
It also means that your mint is running at a loss instead of a profit; making money off literally making money was so common the practice even has a name; Seigniorage.
So while the metals may have an intrinsic value, a new empire, kingdom, oligarchy, or whatever won't see the coins as reserves as metal but reserves of value. As such, all the hard work has already been done. The real effort in making money is deciding how much to make to match the state of the economy, determining how much inflation you want to create to stimulate your economy without stifling it through excessive interest rates, etc. Your subjects, even though your major nation has split, already know how much a loaf of bread or a house costs in terms of the coins that exist, people are familiar with enough with them that counterfeit coins are more easily recogniseable...
Putting all this a different way, changing regime is a lot easier than changing currency and the wise leaders of the fractured enclaves of the former nations will know it. Not to mention the fact that a common currency means that establishing (or even just maintaining the existing) trade relations throughout the fractured nation becomes so much easier. All in all, it may well be a good idea for such a cluster of smaller nations to maintain a common currency, especially if they end up forming any loose confederacy or trade relationship.
There is a price to this kind of thinking, however. It assumes that each of the new nations has a roughly equivalent rate of production, or industry. If they don't, then sharing a common currency is actually a burden to the poorer performing nations.
Think of it this way; one of the reasons that Greece was in so much trouble within the EU was that it didn't have the kind of industry base that Germany or France did. In fact, a lot of their industry focused on tourism. Tourism is a discretionary spend, so when the Euro starts to go up in value against other major currencies on the back of car sales out of Germany and wine and cheese exports by France (massive over-simplification I know) then Greece loses out because the Euro costs more, meaning tourists choose to go somewhere else for their holidays.
If the same thing is true in your fractured nations and there is an imbalance between some of them, expect them to branch off and form their own currency that can change in value against other currencies to protect their industry and export sales. That said, at least one of the states to form out of the former nation is likely to keep the original currency, and as such all nations will respect its value based on the relative prosperity of the nation that uses it.
Put simply, keeping an old currency in circulation even after regime change can be a great shortcut that keeps industry going while the high priority aspects of the regime change can be attended to, and then the currency can be reviewed in detail later on as the time and priorities permit.
