Let's say that for whatever reason, a population of a country has completely lost interest in anything to do with politics. Nobody shows up to political rallies and nobody votes. Everyone prefers to work on their hobbies or go to the pub or watch soaps on TV instead.
Assume this happens in a current-day first world country with a representative democracy. Would the government change into a different system? Would it still have power, or would it become even more powerful?
Politicians would become much more corrupt.
The reason why democracies are usually less corrupt than more autocratic government forms is because democratically elected politicians depend on the public opinion. So when nobody cares anymore what they do, they are free to abuse their power in any way they want. Expect politicians to embezzle huge amounts of tax money.
The corruption will likely tickle down through the whole government apparatus, so don't think you can get anything done without a bribe. Well, you could start a protest movement against the corruption and demand better control of the government officials, but on the other hand, a new episode of World's Funniest Cat Videos is on.
However, this corruption would make the government very inefficient. When the whole government is more concerned with their own good than the public good, it won't get anything done anymore. So you can expect other organizations to unofficially take on government functions.
You want to sue the neighbor but you can't afford to pay the bribe for the judge? Just hire some thugs from your local crime lord and have them beat him up. You need protection but the police chief has a grudge against you? Just hire a private security company. This will empower these organizations and allow them to set up their own rules. So you can expect society to turn into a mafia state where the government is only nominally in power and the force which actually governs daily life are the crime syndicates.
Trotsky is claimed to have said:
"You might not be interested in war, but war is interested in you"
J.R.R. Tolkien expressed a similar sentiment:
“It needs but one foe to breed a war, and those who have not swords can still die upon them.”
In organizational theory, politics is defined as a means of allocating scarce resources. If people are not interested in politics, then they are either not paying attention, or more likely we need to dig a bit deeper.
People who are not interested in party politics might be like the citizens of "People's Republics" in the 20th century. They were corralled to vote in elections where the Party candidate won with 98.9% of the ballots, and knew with utter certainty that their desires counted for nothing in the greater scheme of things. Today in Western liberal democracies, we are seeing something similar. First written about in a book called "The revolt of the elites", people in the political, academic, bureaucratic and crony capitalist class have essentially rigged the system against the rest of the voting public, corralling profits but "socializing" losses (i.e taxpayers absorb the results of bad decisions).
In the case of the People's republics, politics as a means of allocating resources was pushed underground, and black markets, cynical corruption of officials all the way up and down the line and sheer criminality was the rule of the day. One common saying in that era was "They pretend to pay us; we pretend to work". The collapse of communist regimes was accelerated by the hollow economic structure as people worked essentially underground for their own ends, rather than for the State.
In Liberal Democracies, we see an incoherent rage with is expressed in Europe by things like the Brexit, the rise of Nativist parties in major European countries and anti immigration/globalization backlash. (as a footnote, most nativist parties in Europe are incorrectly called right wing by the press; a look at their political platforms as published reveals right away they are far left National Socialist parties)
In the United States, we have the spectacle of anti establishment candidates like Bernie Sanders almost winning the Democrat nomination, Donald Trump wining the Republican nomination and "third parties" like Garry Johnson of the Libertarian Party and Jill Stein of the Green party polling in double digits (in the past, a third party candidate with 2% of the vote was an outstanding result).
Less often spoken of is the erosion of the Rule of Law in the United States and more open corruption by government officials. We also see people being strung along with broken promises, for example State employees expecting a government pension might be in for a shock: Government pensions are estimated to be unfunded by 3 trillion dollars and that is outside of the Federal or State debts.
So a society where people are uninterested in politics due to the real or imagined idea that their preferences are not going to be accounted for will devolve into a lawless society, where social cohesion devotes into family groupings, quasi-criminal organizations and corporations capable of using bribes and other crony capitalist tools to remain afloat. People are still allocating resources, just outside of the official hands of the State. Politics outside of crime will be more like "office politics" rather than party politics.
As a counterpoint to this parade of negativity, if politics is the means of allocating scarce resources, it is theoretically possible to go beyond politics in a post scarcity society. Since social, political and organizational forms will be changed beyond all recognition, this period is sometimes considered a "mundane singularity" (term coined by Brian Wang of NextBigFuture).
Just for fun, add one more variable - what if people took no interest in politics, stopped voting, and also stopped paying taxes?
Every government depends upon legitimacy, namely that people should continue to work, pay taxes, and perform civic duties. Without this legitimacy, people would simply stop obeying any of the laws, and the profound expense of "putting a gun to everyone's head" would lead to an immediate collapse. You can see this in nearly every collapse of a government in history; at the end there is a dramatic increase in taxes (or inflation, like the Zimbabwe collapse) to pay for police forces, but the higher taxes erode public confidence that much faster.
If voting were to cease, a "democratic" government would not have any legitimacy, which would ultimately undermine, not enhance its power. Any policy is seen as being created by a small cabal of individuals who have little connection to ordinary citizens, and as such opposing that policy is no longer "opposing yourself." Imagine a "vote" in which 2% of people vote, the winner carries votes from 1.1% of the population, but 20% of people quit paying taxes and 40% of people are buying guns. There is certainly a "vote" happening, just not at the ballot box.
Further thoughts...
In autocratic regimes legitimacy comes from the idea that governing authorities have some sort of special enlightenment to rule that others do not have. This might be of a religious or non-religious nature, with the latter appearing often in Communist regimes. It boils down to "they know better than I do."
In modern democracies this legitimacy comes from the idea that your vote allows you to have a say in how the government operates, and on the flip side, that all elected officials were chosen by the people, and therefore what they do is really just an expression of what the voters want. So to oppose a democratically elected government is to "oppose yourself."
Where this gets interesting, of course, is when a government goes on the same course regardless of who is elected. This would break the idea that your vote has any effect. Or, if no one votes, a politician cannot claim to represent the people, because "the people" didn't choose him, just the 2-3% of political hacks who make up the parties.
At this point the only winning move is to start behaving more like an autocratic regime, and perhaps start some agencies which give large amounts of funding to academic leaders, who in turn act as cheerleaders for the regime. This would give the regime the appearance of being "enlightened" because all of the "enlightened academics" would be backing them with "scientific proof."
Fans of Adam Curtis' award-winning BBC documentary Century of the Self know that this is currently the case with most Western democracies and has been for quite some time. Without delving too deep in the documentary (even though you really, really should watch it), here's the (very) basic premise:
Edward Bernays - Sigmund Freud's nephew - used his uncle's theories that humans are not rational economic actors and instead primal beings driven by base instincts like fear, lust, hate, jealousy, greed, and so on, as a means to sell products to people who did not need them. This led to the creation of the public relations industry, with impacts such as the idea that women can smoke cigarettes (which used to be explicitly taboo) to be seen as "independent" (read: exactly the way the Philip Morris corporation wanted them to see themselves as they bought their products), as well as what we consider the typical American breakfast ("Wakey wakey eggs and bakey" was a Bernays creation).
Various business and government interests determined the best way to keep people on a perpetual consumerist hamster wheel was to turn them from active citizens - where they seriously research politics and focus on what is needed - to passive consumers, who are placated with shiny things and new cars and politicians who promise the world to get elected.
So, to answer your question, society would look almost exactly as it does today.
When was the last time you personally looked up a political candidates' policy platform?
Exhibit A: The United States in presidential and non-presidential (bold) election years. Source
So. What happens in these off years, when voter turnout is significantly lower?
Well voting in off years tends to be more conservative as older voters are more likely to cast a vote all the time.
So a few notes on your scenario.
The lower the voter turnout the easier it is for some group not in the majority or carrying a majority sentiment on a particular issue to take power.
If literally everyone the government as was ceases to be. You cannot have an elected democratic government if no one votes...I have trouble comprehending a situation where literally no one votes...maybe it could dwindle over time... I suppose the most likely scenarios would be either
The government gets a few voters to for itself and votes to change the laws governing the government to just let them do whatever they want for however long they want.
Corporations step in and take over the responsibilities of governance.
Odds are both scenarios end horribly and with mass oppression, environmental catastrophe and in the end...revolt.
These people's grand children will be spitting on their grandparents' graves.
This state of affairs can come about in two ways.
Everybody has lost hope and no longer think voting can help.
Everybody thinks that the government is doing a good job and feel no need to interfere with it.
The first alternative is grim, as other answers have described.
From a world building perspective, I find the second alternative more interesting. (Even if less likely) The government is doing their job of administrating the country and everybody is content. People are glad somebody is taking on the boring job. Somebody else, that is.
One interesting consequence is that politicians would no longer be celebrities. No TV channel would send political debates since nobody would be watching. No newspaper would print articles about one politician criticizing another for being too soft on some issue. People wouldn't want to read about that, so the paper wouldn't print it.
This would mean that people who just like the spotlight would no longer look for jobs in politics. That would leave the jobs to actual public servants who could then get on with the job of running the country smoothly and painlessly.
Unfortunately the situation wouldn't be stable. There would be corrupt people who would seek out the very real power of a government job. For these people the lack of spotlight would be perfect. They would slowly turn the blessing into a curse, and we would be back with alternative one.
Still, I can dream...
So...you said the population of a country. Weirdly, that includes the politicians THEMSELVES. You did say everybody prefers to work on their hobbies. George W. will continue to paint, Jeb will take up crochet, Hilary will learn woodworking, Trump will stop reading articles about himself and start reading the classics. All the lobbyists will go fishing. No one will run for office because no one will be interested in politics. Therefore, there will be no government. But if people are interested in getting things done, on a local level, they will. It would look something like the direct democracy anarchy system. Of course that would mean they would have to show up to vote, which is an interest in politics. So I guess the people who can afford it will simply hire people to patch up the streets they use.
Hitler opens Mein Kampf with a warning that exactly what he was going to do is what happens when the general population takes no interest in politics. He himself entered politics because he views that the more the general populace ignores politics, the more totalitarian the regime can be.
If I may get up on a soapbox, I dare say that his rally to get the general populace interested in politics is the reason for the banning of Mein Kampf. There is actually very little antisemitism in that book.
Jose Saramago has a book called Seeing. In the book the citizens of the capital of a country does not vote for the election and they do the elections again. They still do not vote and things start to happen. I don't want to give spoilers but let's say it doesn't end well for the citizens.
This is not a complete answer but I can't comment because of my reputation so that's the only way to mention the book.
