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As an American, I have been watching the Brexit proceedings with some confusion. To all appearances, British politicians do not want to compromise, and the European leaders do not want to compromise. This suggests that a no-deal Brexit will occur on the 12th of April (Friday). Given the dire warnings about the dangers of a no-deal Brexit, I would have expected people to be taking precautions, such as stockpiling food or medicine. Why is that not happening? Are the deleterious effects of a no-deal Brexit longer term rather than shorter term? Or is everyone assuming that some kind of deal will be struck at the last minute?

Given the dire warnings about the dangers of a no-deal Brexit, I would have expected people to be taking precautions, such as stockpiling food

This is happening, from the FT (February 2019):

But the closely watched purchasing managers’ index compiled by IHS Markit/CIPS on Friday showed concrete evidence that stockpiling was becoming widespread — particularly in the food and drink, clothing, chemical and plastics, and electrical and electronics sectors — and was being implemented at more larger companies than small ones.

or medicine.

Same here, from the Guardian (first quote, second quote):

The government is in talks with drug companies about funding the extra costs of stockpiling and flying in vital medicines in the event of a no-deal Brexit, the health secretary has revealed.

Patients should consider stockpiling their own drugs if it looks likely the UK will leave the EU with no deal, the pharmaceuticals industry has said, telling MPs that such a scenario could be “catastrophic” for medicine supplies and necessitate emergency powers.

Why is that not happening? Are the deleterious effects of a no-deal Brexit longer term rather than shorter term?

There are many effects, both in the long term and in the short term. The stockpiling is actually only a band-aid solution, especially with regard to fresh food one can only stockpile for a few weeks (or less), according to the BBC:

Fresh food - which cannot be stockpiled - is the biggest concern. A vegetable shortage last year, driven by bad weather in southern Europe, highlighted this dependence, and led to a flurry of pictures on social media of empty supermarket shelves.

The thing with most food (and other importers, like manufacturing plants) coming into the UK is that it works with so-called just-in-time deliveries, from the BBC:

The UK's supply chains work on a "just in time" basis, meaning stock is delivered daily, not stored for long periods in warehouses.

[...]

To illustrate the complexity, Paul Martin, UK head of retail at consultancy KPMG, says 290 trucks come through Dover each day just supplying citrus fruits. He says any level of disruption, such as trucks needing to pass extra inspections, is likely to have an impact.

One of the 'problems' here is that UK imports rely heavily on the Dutch Port of Rotterdam. Indeed, many imports to the UK from Asia go via Rotterdam. Or to quote Mark Dijk, the Port's external affairs manager (as cited by the Independent):

“We are also a hub for the UK. All the deep-sea ships from China are coming into Rotterdam, and their goods are then going into shallow water ships to the UK,”

Or is everyone assuming that some kind of deal will be struck at the last minute?

No, as pointed out by Fizz, there are no-deal preparations going on on both sides of the Channel. And those preparations have been going on for a while now.

The UK government has put out no-deal Brexit info for businesses. (And so has the EU.) But this is presumably not expected to be like a natural disaster, so the general public hasn't been instructed to do anything in particular, insofar, and as far as I know.

For individuals, there are instructions/information dealing with the new border procedures, EU studies programmes, EU family law, and "buying things from Europe".

The "should" part of your question probably only admits answers that are primarily opinion based, at the moment.

The British should certainly be preparing for it, and I think some are, but I would be shocked if the UK actually crashes out with no deal on April 12th. Like everyone else, I'm not entirely sure what is going to happen as things seem to change almost on a daily basis. For example, last week President Macron was signalling that he'd had enough of the UK's indecision and would veto any extension to the leave date. However, almost immediately after that, the Taoiseach of Ireland Leo Varadkar said that whoever pulled such a move would "never be forgiven".

Because of that, I can't see a hard Brexit being precipitated by the EU, because their perceived self strength lies in their unity, which until now has been unshakeable. What's more likely, at least in the short term (Theresa May is asking for an extension to June 30th), is that the proverbial can will continue to be kicked down the road by the British government whilst it remains paralysed by indecision and division.

A hard Brexit can only happen now if a British government allows it to happen. There are still plenty of politicians and people in the country who want that.

The government is between a rock and a hard place.

Full scale preparations would cost a huge amount of money, and cause the public to start panicking. Imagine if the government told supermarkets to stockpile long life foods, people would go crazy buying up supplies.

Some years ago there was a problem with the supply of petrol with a similar reaction. People queuing for hours to put a few litres of fuel in their cars out of fear. People stockpiling and taking advantage with high prices.

On the other hand, if the UK does crash out of the EU it's going to need those stockpiles. The government has been trying to put contingency plans in place, but it's not been going well.

So the UK government is strongly motivated to keep its preparations low key and largely secret (e.g. possible medicine shortages, having the army on standby), but also needs to spend billions on them to ensure it doesn't have mass panic, riots and people dying...

All the while saying that no-deal is a real threat as a negotiating position with the EU and with it's own MPs and Parliament, and simultaneously trying to obey its legal obligation to avoid a no-deal crash.

It's as crazy as it sounds.