That is a nice "just so" story that simply doesn't fit the facts.

Europe had many points in its history where labor was dear. For example wages went up after the Black Death because so many workers had died. Why did the Industrial Revolution not happen then?

Where and when it did happen, the https://en.wikipedia.org/wiki/British_Agricultural_Revolutio... preceded it. Meaning that lots of laborers were displaced from fields (particularly due to enclosures) and forced to move to cities. This lead to an excess of labor in the cities, not a shortage.

The evidence is rather better that the progress of science and technology simply hit an inflection point with the invention of things like the steam engine (factories could be put anywhere) and the spinning jenny (machines could make cloth far cheaper than it had been). In a society which already had a free market bent (it was not an accident that The Wealth of Nations came out of the same time and place), these things were seized upon, imitated, and further innovations proceeded apace.

The spread of the Industrial Revolution afterwards was dependent upon the spread of the necessary skills, the investment of capital, and the building of supply chains. Which meant that what started in England spread to places in closest contact with her. Namely the United Stated and Continental Europe. And then only slowly spread elsewhere.

I've been told (also with no references to cite) that this was also one of the reasons why Rome didn't industrialise, they always had a plentiful supply of slaves, so there was no demand for machinery to replace them.

The causation of the Industrial Revolution is one of those historian's favorites because there isn't a definite "yes" to it, and you can point to origins going as far back as the end of the Roman Empire and the beginning of the Great Migration. Europe lost its old identity as it went into the medieval period - with one large invasion after the next, it was difficult to keep track of anything - and the new nations that formed from the ashes gradually lost their taste for subjugating and extracting tribute from each other, preferring trade competition instead, which is an innovation! If you look at most earlier empires, there's trade, but it isn't in a systemically central position because the ruling class would stomp out competition. That there were multiple of these nations is critical to the process: it allowed someone who angered a so-and-so to flee elsewhere and keep working. Fleeing is a common recurring theme in biographies of notable early modern people. This allowed Europe to keep holding on to ideas and discoveries over time, putting it in a good position to absorb the world's knowledge.

Why Britain specifically is a subset of the question. One of the precipitating factors often cited is the enclosure movement, which was part of a push for stronger, more logical systems of property rights(a big philosophical movement originating in the late 1600's and gaining steam in the 1700's - "life, liberty, property", utilitarianism, and related ideas all come from this period, and in Britain these ideas were especially popular). Enclosure had the effect of kicking peasants off the farm, leading them into wage labor, and Britain had a colonial system already in place, giving it many export markets and therefore demand for manufacturing. Add to that the local availability of coal and the gradual ceding of governing power to Parliament(another centuries-old process) and you have something resembling a thesis for preconditions of industrial capitalism - a stable country ruled by its merchants, with a strong export market, availability of labor and energy sources, and sufficient knowledge to usefully organize the labor.

Thank you

The Industrial Revolution happened on England in a time when labor was plentiful relative to the usual rate. There was a run away into the cities happening.