If it's gold or silver (or any metal really) it's still got commodity value. Now the Romans had heavily debased their currency by the time the Western Empire collapsed (they didn't seem to grasp the economic consequences of this) so the actual commodity value of the currency did fall during the last centuries of Western Rome's era.

So when Rome fell, those with currency would have liquidized (literally) their assets and extracted the valuable commodity metals from the currency who's face value had ceased to have any meaning at all and then used the metal itself as a medium of exchange.

Of course, it then ends up being traded and reused across the world, some going to make an armring in Scandinavia, or goblets for the court of Charlamagne, with some being remade into coinage within the area of the old Western Roman Empire (mostly crudely struck until minting quality improved again) or even being made into currency in the Eastern Empire and beyond.

The thing was, unlike modern fiat currency, metals were held to have some sort of intrinsic value so metal from these coins could end up reused ad-infinitum.

As for your question as to after the Empire collapsed, think about older coins today. Their face value might be say, a British gold guinea from the 1770s has a face value of £1 1 shilling (£1.05 in decriminalized currency) however, it contained 7.7g of gold. Gold today (and it changes day to day) is worth £24.83 a gram, so a guinea with 7.7g of gold in it is worth £191.19 in today's money.

The gold guinea is obsolete and yet is worth massively more than it's face value, so the value of the gold, silver, copper etc in the coin would likely dictate them having any monetary value in a system which had essentially regressed from currency based (i.e. the majority of Western Europe immediately after the fall of the Roman Empire) to simple barter in most places.

Greek, Roman, and Byzantine coins made from base metals are still widely available today if you're curious. My local coin and jewelry shop has a big dish of them for you to root through and buy at $3 a piece. The clerk told me that every old-time farmer in Europe has a jar of these from plowing, and they periodically sell them off. I don't know that that's true, but it sure sounds plausible. I've got a couple in my wallet, just as curiosity pieces. (Someday I'm going to win a bar bet abut who has the oldest object on their person.)