Paper currency is often used after the government backing it collapses.

Not only that, people will use anything as currency in the absence of fiat. Cigarettes, tea bags, polished rocks, etc.

Are miners still going to be operating in this post-collapse environment?

Whoever operates is going to become rich. So someone will step up to satisfy the need.

You need a hash rate at least vaguely close to the pre-collapse hash rate, or else the network will grind to a halt. Will they be physically capable of that?

This assumes a sudden collapse event happening within the 10 minutes that it takes to reset the hash rate. I'm struggling to think of global scenarios where that can happen so quickly.

What are the chances that collapse is a sudden event vs a cascading deterioration event?

It takes two weeks to reset the hash rate, if the hash rate stays constant. If it starts dropping, the time goes up.

can you explain this further?

Sure! The difficulty is supposed to adjust approximately every two weeks, and it readjusts based on how quickly blocks were found after the previous readjustment. The target is for blocks to take an average of 10 minutes.

The trick is that the difficulty adjustment time is not actually measured in weeks (or seconds), but in blocks. Since blocks are supposed to take 10 minutes, the difficulty readjusts every 2016 blocks. It looks at the time needed to complete the last 2016 blocks, and readjusts so that the next 2016 blocks will take two weeks at that rate.

If the hash rate is steady, then that readjustment will happen close to the two-week mark, and of course it will do very little because the rate is steady.

If the hash rate is increasing, then the period becomes smaller. As a simplified example, consider what happens if the hash rate is constant for the first week of an adjustment period, and then doubles. The first 1008 blocks took a week. The next 1008 blocks will take about 3.5 days. The next readjustment will therefore happen after about 10.5 days, not 14 days. The real Bitcoin network looks a little bit like this: over its lifetime, the actual difficulty readjustment period has averaged 13.17 days, not 14 days, because of the steadily increasing hash rate.

Now consider the opposite case, where the hash rate is cut in half after a week. The second batch of 1008 blocks will take two weeks, for a total of three weeks.

Worse, consider a case where the hash rate is cut in half every week. You'll (probably) never reach the next difficulty readjustment, and blocks will take longer and longer to mine until they're taking days or weeks instead of just 10 minutes. You could get into a spiral where a single event that significantly cuts the hash rate results in the network becoming much slower and less useful, so people use it less, so miners earn less, so they drop out and the hash rate drops further, etc.

It's hard to say just how the real-world hash rate would change in response to a catastrophe, of course. Maybe that sort of shock wouldn't happen. But this is the general idea of why Bitcoin could run into trouble due to a quick but not quite immediately apocalyptic decrease in the hash rate.