Short term isn't an issue, other than the short term inconveniences, and crash of many different markets.
Bitcoin would be ok in market terms immediately because no one would have had the ability to sell it.
After the internet came online, bitcoin would fall with the rest of the market based off of market fears.
The bitcoin network would correct itself immediately with the longest proof of work.
There might be issues if there is longer than two hours between miner's clock though..... but even then, the longest chain wins.
But, like shut down for a long term?
- One country shutting down its own connection to the rest of the world
As pooya says, there would be a leak of data somewhere, so this would be pretty unlikely, but I imagine a few things could happen:
Both groups continue on as normal. Two forks.
The smaller group might keep track of all transactions (meaning, more than what's in the blockchain but the messages that are sent to be put into blockchain) for when the groups join up again. However, the smaller group of countries would soon implement a hard fork with a small change to invalidate the larger group's proof of work to not lose their transaction history when they join up again. The groups could continue isolated for decades. The much smaller group would have an incentive to force their own currencies and block all transactions that didn't originate during their isolation to not be blown out of the water when the groups join up again. It would be a separate coin for them.
If people in the smaller group were also clever enough, they could keep the keys to all of their transactions to be used twice: in their fork, and in the larger group. The larger group continues with mining power and use, but the smaller group despite their lack of mining power has effectively older addresses and keys. They can still transact on the longer, larger group's chain without their new-isolation transaction data having been saved on it. So, the smaller group would suffer short term but would have some benefit later on that could balance out. The larger group wants to maintain mining power so that they don't lose their transaction history, and because they are larger, they can't determine which transactions are only from the smaller group.
These separate pressures between the two groups could force them to reconcile sooner.
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[warning! muddled paragraph:]
For the armaggedon scenario, which is more unlikely, you already have worse things to worry about, so let's ignore that. When the internet came back online again after .. months? years? decades?, people would want to have a unit of accounting that they could agree on but didn't have to agree on. Enter bitcoin. If the internet was being restored very slowly, you could start anew with the mining process and new genesis block, which would work to rejoin a global market. It depends then though whether the people who had mining rigs were trusted enough to be given the most amount of control over the currency. If miners were trusted enough and it was a slow recovery, then mining would start instead with new genesis and not the current database. If miners weren't trusted, the old database would suffice on the premise of it being reasonably distributed globally. Not perfect, but probably better than whatever other markets that were now valueless. It would be simplest to use the currently distributed database of values globally because no one would agree with simply giving everyone a baseline amount, because I don't think distribution is possible to do again without manipulation. You'd probably have a lot of alt coins enter the discussion too.