Let’s say that Russia invaded Poland.
Poland does not use the euro; it uses the zloty, its own currency. A war would likely cause a severe recession and loss of faith in the domestic currency. What measures is a nation likely to take to prop it up?
Would they ban conversion to foreign currencies, thus forcing Poles to have to keep their currency as is (no conversion to euros or dollars, for instance?)
That would not prop up the currency, if by “propping up” you mean stabilising the exchange rate. An exchange rate requires that the currency is being bought and sold - the exchange rate is simply the price at which such buying and selling takes place. If you bring all of that to a standstill, then there wouldn’t be any exchange rate. Sure, you could post an “official” exchange rate, but if no actual transactions take place (at this rate or any other), then that would be a meaningless figure.
Most likely there would be convertibility restrictions, but subject to exemptions in the sense that for certain transactions (such as the purchase of essential goods or equipment abroad by domestic buyers), foreign currency could still be obtained at some defined official exchange rate. Most likely, the counterparty to such a transaction, i.e. the one who sells euro or dollar in exchange for zloty, would not be a commercial actor on the open market (they would be unwilling to buy a currency that’s subject to such restrictions) but rather a governmental actor such as a central bank, and the euros or dollars would come out of the national foreign exchange reserves.
Similar systems were in place for a long time in the Eastern bloc until the early 1990s; if a domestic company needed to buy goods from abroad, or a resident of the country wanted to travel to a Western country for an approved voyage, and foreign currency was needed for that, then they could buy a limited quantity from the central bank in exchange for domestic currency at an official rate. On the black market, the rates at which foreign currencies were (illegally) traded against the domestic one were very different from official rates. No doubt, in your scenario a similar black market would soon spring up.
Governments have been doing similar things for centuries. You don’t need an invasion.
Are we talking about a government fighting a war on its eastern front, or an occupying force trying to keep the economy (and hence the currency) alive?
In a war footing, the government will do anything and everything to ensure it has the men and materiel needed for survival; hence actions like invoking the War Powers Act to force industry to supply it with what it needs, from boots and uniforms to armored vehicles. In those cases, the government pays what it wants in its own currency, and the recipients have no choice. People are paid in this currency and must buy their daily essentials. The law of diminishing expectations may result in a form of inflation - “It may be that in 6 months this stuff is worthless, so I better sell my bread and sausages for a lot now, so I can buy that car before the currency collapses.” Plus, the temptation for the government is to “borrow” (print) money to finance the war, more so than worrying about longer term currency stability. Who cares if the Ukrainian Ruble is stable if you as the ex-president are sitting in a Siberian gulag? Usually the government invokes progressively more authoritarian measures as the economy slides into chaos - see the current state of the Venezuelan economy. Limit exchange rates, forbid large conversions to foreign currency, set prices for food and essentials, prosecute hoarding against those who refuse to sell at the mandated prices. it’s more of a downward spiral, but then the country is fighting for its life. Inflation is the same anywhere - if you print too much money to chase too few goods, prices go up. The usual incompetent government answers, force compliance and print even more money, don’t actually fix the problem.
The biggest issue would be foreign exchange; the German and American arms dealers are not going to accept Zlotys, they want Euros or Dollars. The government will take whatever measures or seize whatever it can to pay for the next load of guns and ammo. Its options are limited if the merchants have limited faith in its ability to win the war - only something that can be loaded onto a ship and moved out of there is acceptable.
Meanwhile, the question of confidences in banks is an issue. In the days of gold standard, who held the gold and how safe it was determined the stability of banks. Today, the question would be- what security does the bank have, and should the country lose, how likely the occupying power would honor that? A mortgage on the Polsky Vodka Works would be valueless if the occupying power seized it and declared all pervious obligations null and void. (OTOH, when Allende seized copper mines from American owners in the 1970’s, they went all over the world tying up copper shipments in court over lawsuits that the nationalization was a theft and demanding ownership of the copper - effectively cutting off any hard currency revenue from the mines. The economy is always a two-way street)
People may lose confidence and create a run on the banks, take out their money and buy things they think will have value if the zloty is no good - jewelry, durable goods, large amounts of food, even foreign currency at ridiculous (black market) exchange rates. Governments in economic crisis will also often impose strict limits on withdrawal amounts to prevent them having to prop up every bank.
If a government is winning or (more likely) manages to grind down to a stalemate, then it can worry about the economy. First priority, that people are fed; then worry that they have other needs. then worry about longer term stability. Announcing that banks will be supported by the government, ensuring less economic disruption so a decent flow of consumer goods continues, etc.
the biggest problem an occupying power may have to deal with - besides insurgency disrupting the economy - is what to do about the economy. Replacing a currency is a huge job; and likely the defeated outgoing government along with general chaos may have left a legacy of inflation to be dealt with. Plus, chasing down exactly what happened in the absence of likely destroyed records - the occupiers want to be sure no cronies walk away rich, retrieve any moneys transferred to keep ex-presidents in luxury in Argentina, repudiate debts to whoever sold arms to the defeated government, etc. the first step to stability, as always, is to ensure there are goods for sale at a reasonable price, especially to start with - food. Get the standard industries back working as fast as possible, which means fixing any supply chain issues -which means guaranteeing any foreign debts related to the supply chain, etc.
Short answer is always - the economy will take a second fiddle to the war until it’s over. the war, that is…