How long would it take for the earth's crust to move in such a way that we, as a species, will be forced to migrate to other different lands than the ones we inhabit today? The result being that our ancient cities likes London, New-York or Athènes will be found under sea level. Could this happen?
New York is 33ft above sea level. London is about 135ft, and Athens highest point is over 1100ft but generally around 235ft above sea level. So presumably we are talking very different time frames for each city, but to directly answer your question: No, possibly never.
Sea levels rise and drop because of global sea temperature and the amount of polar ice that locks the planet's water above sea level. Earth has gone through cycles of heating and cooling, which has caused the polar ice caps and glaciers to grow and shrink. The last interglacial period when Earth was only a little warmer than it is now, 125,000 years ago, and sea level was estimated to be about 12 to 20ft higher probably due to the melting of Greenland and the West Antarctic ice caps. New York would have considerably less land, but London and Athens would not be under water.
In contrast, during the last glacial maximum about 19,000 years ago, sea levels dropped by over 400ft exposing the continental shelf and creating land bridges that connected continents.
If you want the movement of the Earth's crust to actually sink the cities below sea level, there are several factors to consider. The Earth's crust is technically floating on the mantle, so land can be driven upwards or sink downwards depending on what is happening to neighboring areas under the crust. We all understand that mountains are uplifted when two tectonic plates push together, but the weight of glaciers and the polar ice caps is also considerable. The heavy ice pushes down on the crust beneath it, and part of the mantle "squishes out" to nearby areas to lift the land up. As the ice melts the mantle flows back under the glacier and the surrounding lands sink – called "glacial rebound". The oceans are also estimated to weigh considerably and push down on the sea floor, rising continents higher. All of this makes estimating the "true" sea level at various points in Earth's timeline very difficult.
Another factor is the ice caps don't simply melt and the global sea level rises evenly all at once. Water is trapped behind ice walls for thousands of years like North America's Lake Agassiz. Eventually the ice wall melts causing outburst floods, but even these can take hundreds of years to drain into the oceans. The Zanclean flood which filled the Mediterranean Sea from the Atlantic Ocean over 5 million years ago might have happened quickly or slowly, we really don't know. Some scientists postulate the Mediterranean filled at 30ft a day with a dramatic 1 kilometer waterfall, but others believe it was gradual around 10,000 years.
Continents are also sliding around, and the next Pangea-like super continent is estimated at 250 million years from now. Where today's cities end up, along an ocean coast or not, is really only a guess.
So it makes for a dramatic mythology, but entire civilizations sinking beneath the sea is not a world-wide event. If you want a reason for a specific coastal city to sink, it is possible with local volcanic activity, and has happened many times within written history, for example: the Roman city of Baiae.
If you want this to happen in your world, then yes it will happen. If you don't want it to happen, it won't. It's that simple.
Here at WorldBuilding, we don't discuss things which are happening in the real world. We do follow real life laws of science though (if you want that).
So, considering that currently global warming is a much more serious threat than continental drift, it is indeed possible that shoreline cities would get submerged if we continue to burn fossil fuels at the current rate. Records show that CO$_2$ levels have always been higher than our current levels, in all of Earth's geological history. Records also show that sea levels have always been higher than our current sea levels, during most of Earth's geological history (all of geological history, except the glaciation times). If we go on to double out carbon content in the atmosphere, we will quickly face a rapid rise in sea level and the historical cities will be submerged.
Currently it is impossible to know exactly how long it would take to double our current atmospheric carbon content. There are global efforts to cut carbon emissions, and then there are people like Trump ... Some models suggest that burning fossil fuels at current rates will raise sea levels by ~30 feet in the next 160 years or so.
However, also keep in mind that rising sea levels are not guaranteed to submerge coastal cities. First world nations are actively building sea-walls to hold off ocean water from their coasts at distances of up to 5 miles.
So, all in all, it all depends on what you want it to be. If you want sea levels to rise and drown coastal cities, keep carbon emissions increasing in your world and don't let the prosperous nations build sea-walls.
As a semi-plausible hypothesis: Due to global warming the ice sheets of both Antarctica and Greenland let loose. They are skating on a water film, and instead of plodding at usual glacial pace, speed up to a mile a day. This puts some large percentage of the world's ice afloat in a few years.
The net rise in ocean levels is about ~200 feet, about 1/10 of that from Greenland.
Increase plausibility: Only half the ice moves this fast initially. This gives you a 100 foot rise.
This takes out every port in the world, and requires some 90% of the world's population to find a new address.
Problems with junk that gets buried too. Landfills, warehouses with toxic waste.
This is water level rising at something like 10 feet a year.