So basically the argument goes that since many British people migrated to the America in the 18th century, the country experienced a worker shortage. This resulted in a significant increase in labor cost. Since the labor cost rose, people were forced to adopt new experimental manufacturing techniques, to maximize the efficiency of workers and reduce their number to cut the costs.
On the other hand, places like China did not experience any mass migration, and so the cost of labor was low. No point in experimenting with some new technologies when a bunch of peasants can do the work for pennies. And as such Great Britain was the first to experience industrialization.
Is there any credence to this argument? Did Great Britain suffer from a worker shortage in the 18th century, and if so, did it have any influence in starting the industrial revolution?
There are some parts of this that fit with the literature on industrialisation, and some parts that don't.
Allen's factor prices argument for industrialisation suggests that the development and application of labour-saving technology is a matter of relative factor prices, as you suggest. Expensive labour (relative to the price of capital) incentivises substitution. That part is probably correct, though it is damnably difficult to isolate that effect from everything else going on simultaneously. The quality of our wage data is not great, and the builders "day wage" series that Allen uses has been subjected to some serious critique. But I think the overall argument holds, that at least one reason for British precocity was a combination of relatively high wages and relatively low cost of capital. Countries that roughly fit this pattern tend to industrialise faster than countries that don't. China, as you say, did not begin to industrialize until the early 20th century, and even then, only in the most developed, highest wage areas.
Are British high wages a result of a labour shortage caused by mass migration? No. How do we know? Two reasons. 1) Because the vast majority of UK outwards migration occurred in the mid-19th century, after the beginning of industrialisation, not in the 18th century. 2) Because rapid population growth was increasing rather than decreasing the overall supply of workers.
Migration before the defeat of Napoleon was relatively modest. Costs of transport were high, and incomes were low. This meant that accumulating the savings necessary to purchase passage to the new world and sustain oneself while finding employment was difficult. For relatively poor people, the usual solution was indenture. This solved both of the above problems, but came at the steep cost of one's freedom for the years of the contract. Even as late as the 1820s, only 15,000 or so people voluntarily migrated per year from all of Europe to the Americas. African slaves brought across the Atlantic outnumbered free migrants by about three to one. This is simply not a large enough movement of workers to put serious upwards pressure on wages, though it no doubt had some small effect. The real mass migration of workers begins in the 1840s with the decline in the cost of transportation and the slow increase in wages. But by this point, the first industrial revolution was almost complete. Insofar as there is a relationship for Britain between wages and migration, it mostly goes the other way around: higher wages allowed potential British migrants to overcome the cost of migration earlier than migrants from other countries.
Wages were high in the UK because productivity was high. By the late 18th century, Britain was one of the most urban, least agricultural societies in the world. British workers, especially in the North, were relatively healthy, well-fed, and skilled, at least according to Kelly, O'Grada and Mokyr. Canals, coastal shipping and road building had integrated markets and created large domestic markets for artisanal manufactured goods. Coal was increasingly used as a replacement for wood for heating. Agricultural innovation both mechanical and biological had increased land productivity, requiring fewer workers to sustain cities. International trade, including but not limited to colonial trade, brought in revenue that sustained merchants, shipbuilders, bureaucrats, and other producers of urban goods and services. The beginnings of banking and financial services allowed for more efficient commerce, as well as private and sovereign borrowing. The British mercantilist state could harness enormous revenues (by the standards of the time) to explore the world, to fight wars, to monopolise trade, all of which pushed wages upwards. Scientific societies were pushing back the boundaries of our understanding of physics and chemistry. And so on and so forth.
Population growth also does not stop in this period. Indeed, the population of the UK grows quite rapidly across the 18th century, from about 6.5 million to over 10.5 million in 1800. It would be strange to describe the result as a labour shortage, at least in the sense of "the number of available labourers was declining, so wages rose." The correct version is more like "wages continued to slowly rise, despite population increase."