I, for one, welcome our new BitCoin overlords, and would like to remind them that I can be helpful in rounding up others to toil in their crypto mining pools.
That's only slightly glib, in that what you're asking about has happened to varying degrees in various countries around the world, both historically and in modern times. Your scenario is fundamentally the same as other historical financial system and governmental collapses, only with the slight variation that it's an electronic financial system.
The classic textbook example of a total financial collapse is hyperinflation in the Weimar Republic during the interwar period. The Weimar Germans had it especially bad because they really didn't have any options other than using the worthless paper currency, which by the end was being burned for heat, used as wallpaper, and generally used as anything but currency.
The best modern example of a total financial system collapse is Zimbabwe, 1990's through 2015. (Zimbabwe's peak month of inflation is estimated at 79.6 billion percent in mid-November 2008, which works out to 30,700% inflation per second), and they still... exist, and in largely the same horrible mess they've always been in. A far cry from total chaos or a societal collapse. When a financial system collapses, people just use something else as currency, and ultimately the result is that the government-issued currency becomes completely useless, is not used, is no longer printed, and people standardize on something or somethings else to use as currency. Maybe, long term, the government is able to re-establish government-backed fiat currency, but if not, and until they do, people rely on other nations' currencies, and traditional "value stores" like gold.
The current situation in Argentina, post-Chavez is instructive, as it is likely similar to what we could expect from your scenario for any given society, not counting any international economic repercussions or fallout. Bad, painful, but far from total chaos. Society doesn't collapse. Argentinians now are using currencies from other countries and commodities like gold as currency, rather than untrusted, mostly useless state currency. (Even with the government largely criminalizing the use of foreign currency and mandating obscenely bad official exchange rates.)
Someone or something destroys the financial world every 10 to 15 years, but we recover and carry on every time anyway. Most recently, it was the sub-prime mortgage fiasco that wiped out tens of trillions of dollars from the US economy. Before that it was the dot com bubble bursting. (Twice, actually, late 90's and again in the early 2000's). Before that it was the banking collapse in the 80's, before that the OPEC embargo of the 70's, and so on, probably going all the way back to the beginning of history or so.
Every decade or so the world gets destroyed, and every time we recover, rebuild, and prepare to do it all over again.
So, let's pick the US for our total financial collapse, as it's probably the worst case for global implications and repercussions as far as how financial collapse in a single country would ripple out to the rest of the world. Frankly, it doesn't make much difference how it happens... whether it's hackers, war or invisible pink unicorns, recovery looks basically the same, and it's basically the same thing that happened when banksters destroyed the world 8 years ago, dot com companies and moron financial analysts destroyed it before that, and so on.
In what ended up being massively longer than I'd like, you can see my analysis of what implications a US financial system collapse would have, below, and there's simply nothing there that's unprecedented. It's all stuff we've survived in the past, so I see no reason that we wouldn't survive it again.
US dollars become worthless. This is brutally painful for Americans and wipes out tens of trillions of dollars worth of wealth in America alone, the American economy sinks further and faster than it did in the Great Depression. Within the US, this creates a societal reordering, rather than a societal collapse. The current wealthy elite whose fortunes are based on the newly worthless US financial system and fiat currency go from billionaire to broke overnight. (Just as happened in the Great Depression and the Black Tuesday 1929 stock market crash.) People fortunate enough to have substantial holdings of what replaces USD become the new wealthy elite. At this time, replacements for USD would likely be precious metals commodities (gold in particular), GBP or Euros, and people with significant numbers of blockchain-based crypto currency (BitCoins, LiteCoins, etc). The existence and current semi-accepted state of crypto currency is a new phenomenon, yet it's all but guaranteed that crypto currency will be a significantly dominant replacement for whatever government-backed currency gets wiped out in any future financial collapse - it has all the advantages of government-backed, fiat currency (and some significant advantages fiat currency doesn't), but most significantly for a financial collapse situation, is completely unrelated to the current financial system or government fiat currency.
Countries that are major global trading partners with the US suffer severe economic destruction as collateral damage from the US financial collapse, of course. Countries who use US dollars as a defacto currency will be similarly devastated, and find themselves turning to similar replacement currencies as seen in the US. Countries that peg their currencies to a rate based on US currencies experience a brief and unpleasant currency correction shock, but are not too bad off. Even the countries that artificially suppresses or elevate their own currency evaluation via this USD linkage don't suffer significant long-term repercussions, since they just switch to pegging against the Euro instead.
China would end up suffering almost as much economic damage as America in the most optimistic analysis, with a significant chance of ending up suffering even more from a US financial system collapse than the US would. China owns massive amounts of American private and sovereign bonds (debt), which is now all gone. Use of monetary policy and a pegging of the Yuan to USD at an artificially low rate has historically created large economic growth based on exports to the US, at a cost of making the Chinese economy dependent on those exports. The US financial collapse wipes out US demand for Chinese goods, dragging down the Chinese economy with it. The massive loss of wealth for both Chinese companies and government from newly worthless US debt causes panicked stock sales on a large scale. The Chinese stock market is currently heavily artificially inflated by the Chinese state, which no longer has the ability to continue the charade. The Chinese suffer a market crash as bad as, or worse than, the Black Tuesday market crash of 1929 in the US. Existing concerns about overstatement of Chinese economic growth by their state agencies amplify the stock market crash and result in Chinese economic contraction on a scale not seen in major economies since the Great Depression. State controls on the Chinese economy go into overdrive, in an attempt to prevent total economic collapse, but fail, because the foundations of the Chinese economy (ownership of US debt, exports to the US at highly favorable currency rates, an artificially inflated stock market) have been ripped down by the US financial collapse. There is a significant possibility that expensive, mostly empty cities built by state planners will need to emptied back into the rural areas as a result of the economic collapse, in a reverse-flow replay of the Mao-era "Great Leaps Forward". This would have the effect of collapsing real estate prices, which would further exacerbate the economic and stock market crashes. China likely has no possible short term solution or mitigation to their economic implosion.
The EU and non-EU European countries go into economic depression, but not devastatingly so. Like everyone else, China and the US comprise a significant part of their economies, but not enough to risk financial or societal collapse as a result. Other "1st world"/"industrialized"/well-off countries outside of the EU (South Korea, Japan, Australia, etc.) end up in a similar boat as the EU, with varying levels of economic depression and wealth destruction as a result of the collapses in the US and China that comprise a significant part of their economies. All these economies will be growing again inside 2 or 3 years.
India gets hit a bit worse in absolute terms than the EU and "industrialized" countries, due to their industrialization efforts being financed significantly by the US and China, as well as the loss of those markets for their good and services. In relative terms, the collapse of China is advantageous to them. India supplants China as the world's largest, fastest growing "developing" economy. With China's economy in ruins, India is now the country that will emerge as the world's next military and economic superpower. Brazil is in a similar position to India economically. In absolute terms, things are a bit worse for them, in relative terms, China's fall means that Brazil joins India and Japan in the top 5 world economies, when counting the EU as a single economy. (EU - #1, US - #2 probably, #3 possibly, Japan - #2 or #3, India - #4 or #5, Brazil #5 or #4. China probably drops to #6, followed by Russia in #7.)
Africa, Russia, and Central and South America are hit hard by the collapse of China and/or America, and the resulting loss of trading partners, investment dollars and/or foreign aid/financial loans, but they're all economic disaster areas anyway, so relatively speaking, the world doesn't look too much different to them.
The impact on the Middle East and the OPEC oil producing countries is highly variable, depending on which specific country you look at. Oil is still in high demand. The devastated US and Chinese and depressed EU economy sees global demand for oil drop 20 to 35%, but this is not crippling to most OPEC countries. Middle Eastern countries that are warzones or otherwise have no functional government control remain as disaster zones. ISIS likely tries expand, forcing direct military action by Israel and Turkey, in conjunction with any governments currently left standing in the area. Similar dynamic for the other Islamist groups operating in failed-state regions of the Middle East. As the two stable powers remaining in the region, Israel and Turkey likely expand their political and military influence accordingly. The House of Saud falls to internal parties or external forces who set up local puppets, as the US is no longer able to prop them up. Probable fractured-state or failed-state dynamic in the Arabian peninsula as a result.
Decreased global oil demand hurts Iran more than most. With US and European
influence and force projection capabilities in the region severely
hampered, there a strong possibility of multiple flashpoints between
Iran and any standing Sunni Arab states, Iran and Israel as well as
Iran and Pakistan. If so, India looks to exert influence on its
periphery against Iran, possibly warming Indian-Pakistani relations.
North Korea collapses with the loss of their Chinese patron,
resulting in several million starvation deaths in the first year.
Korean unification under South Korea likely, after bloody
failed-state dynamic in North Korea, as DPRK military fails to maintain control with Chinese patronage. Probable cooperation between South Korea and Japan substantially warms relations between the two.
So... a lot people are ruined, a lot of people die, some countries get wiped off the map or split into pieces. Alliances shift, wars and insurrections occur. Underneath it all, normal people keep doing what they have to do as best they can do it to survive and put food on their table. AKA, the whole history of the world since the dawn of time or so.
