Would pollution really take a lot of years to disappear from the atmosphere once people stop producing them?
[I apologise for this long &amp; somewhat scrappy answer.  It may contain significant errors, for which I apologise as well.]

How long something persists in the atmosphere once it's there depends, obviously, on how it gets removed from the atmosphere and also if anything is adding it to the atmosphere.  Different things we might consider pollutants will have different lifetimes, often very different, and it's confusing to lump them together as 'pollution'.

However there is a very important distinction to be made:


one thing we could care about is 'how long do we expect an individual atom/molecule/particle of $x$ to remain in the atmosphere before it leaves it?'
another thing we might care about is 'how should we expect the concentration of $x$ to behave in the atmosphere over time?'


These two things are not the same.  They are not the same because, even considering only 'natural' processes (which we might define as 'processes not resulting from the burning of fossil fuels or industrial activity') absorption is not all that happens: emission also happens.

The thing that is interesting to know is generally the second: 'how does the concentration of $x$ go over time?'  That's because no-one actually cares whether a given unit of something stays in the atmosphere for long: they care about how much of it there is in the atmosphere: I don't care whether I'm breathing some particular oxygen molecules, I care that there is enough oxygen in the atmosphere that I can breathe.

Removal mechanisms

There are at least three mechanisms for removing something from the atmosphere.


It may not be stable in the atmosphere.  A good example of this is methane, which gets turned into other things over a few years.  If methane emissions (including natural ones) stopped it would essentially all disappear within a few decades at most.
It may leave the atmosphere by falling out of it or by being involved in precipitation.  This is true for aerosols (particulates) in the atmosphere: aerosols with large particle sizes really fall out of the atmosphere, aerosols with very small particle sizes get removed by precipitation and intermediate size ones experience both processes.
It may get actively absorbed by something on the surface.


A fourth mechanism is 'it may escape to space' but this is not a significant process anything interesting here, so I'll ignore it.  There may be other mechanisms I have not remembered.

Addition mechanisms

There are, I think, two significant mechanisms for creating stuff in the atmosphere:


it may be emitted by something on the surface;
it may be created by some process in the atmosphere.


(And again, it may arrive from space, but I'll ignore that.)

How long things persist in the atmosphere

So, now, how long things persist in the atmosphere depends on what they are: the term 'pollution' bundles together things which have radically different characteristics.


Human-generated aerosols (smoke, other aerosols) have relatively short lifetimes in the atmosphere and are not created by processes in the atmosphere, and so will fairly quickly decline in concentration once emission stops.
Aerosols of water – clouds – are created in the atmosphere, from water vapour which will continue to be emitted from the surface, and are also removed from the atmosphere as rain, but these processes balance out over time and so clouds will persist in the atmosphere.  Clouds are not a pollutant of course: I just wanted to add them as another type of aerosol with another lifecycle in the atmosphere.
$\mathrm{NO_x}$ has a relatively short lifetime in the atmosphere as it's not stable: its lifetime in the lower atmosphere is hours, and in the upper is not more than few weeks.  Some $\mathrm{NO_x}$ is created in the atmosphere by lightning, but I think the quantities are rather small compared to the amounts created by human emissions.  So the amount in the atmosphere will decline pretty quickly once emission from the surface stops.
and so on.


The elephant in the room: $\mathrm{CO_2}$

$\mathrm{CO_2}$:


is pretty much chemically stable and so does not decay in the atmosphere (it does undergo processes of being dissolved in water droplets and then evaporating from them, and perhaps others);
is not an aerosol and so does not leave the atmosphere by the mechanisms which affect aerosols, unless it is dissolved in aerosols of water;
is absorbed by several processes on the surface;
is also emitted by several natural processes on the surface.


The things which absorb $\mathrm{CO_2}$ on the surface are, among others (not necessarily in order):


plants, which absorb $\mathrm{CO_2}$ and turn it into plant and oxygen;
the oceans, which dissolve $\mathrm{CO_2}$;
rock weathering;
other things.


The natural (see above definition!) processes which emit $\mathrm{CO_2}$ are (not in order):


plants and animals which respire;
plants (and animals) which die &amp; decay or catch fire, releasing $\mathrm{CO_2}$ back to the atmosphere;
the oceans, which release dissolved $\mathrm{CO_2}$ back to the atmosphere;
vulcanism;
other things.


So in order to know how $\mathrm{CO_2}$ behaves over time we need to understand how the various processes work: how fast is $\mathrm{CO_2}$ cycled through the atmosphere and how fast does the concentration change.

Understanding this is not simple, and it's why people write big climate and earth system models.  But it's possible to give ballpark numbers.

The answer to the first question: 'how long does an individual $\mathrm{CO_2}$ molecule spend in the atmosphere' is 'a few years'.

The answer to the second question: 'how long would it take for $\mathrm{CO_2}$ concentrations to decline to preindustrial levels if fossil-fuel burning stopped, is 'from a few hundred to a few thousand years'.

The reason for the large difference between these numbers is that the natural processes that absorb and emit $\mathrm{CO_2}$ are very close to being in balance.  This is easy to see because we know that the concentration of $\mathrm{CO_2}$ in the atmosphere was not zero before we started burning fossil fuels.

Just to give some figures (these are not meant to be accurate):


plants absorb about $120\,\mathrm{GtC/y}$ through photosynthesis;
plants emit about $60\,\mathrm{GtC/y}$ through respiration;
about $60\,\mathrm{GtC/y}$ is emitted from decay of plant matter;
the oceans absorb about $90\,\mathrm{GtC/y}$;
the oceans emit about $90\,\mathrm{GtC/y}$.


As it stands this is completely balanced.  Well, these figures are approximate, and there are other processes both of absorption and emission.  But it's close to being in balance.

This is why the concentration of $\mathrm{CO_2}$ would fall only rather slowly if fossil-fuel burning stopped.