. . . [T]he timing of industrialization explains 42% of the variation in GDP per capita today. Likewise, the industrial transition goes hand in hand with the demographic transition for most countries. We find signs of technological catch-up; the later industrializers industrialize at a faster rate and with higher levels of growth in GDP per capita and schooling than the countries that industrialized a century ago. (p. 29)That is form the conclusions of this paper by Jeanet Sinding Bentzen, Nicolai Kaarsen, & Asger Moll Wingender.
Regarding institutions the authors argue that industrialization happens at different levels of democracy, but mainly when the polity2-index is close to full democracy of full autocracy (see figure 10). It might also be possible, they argue, that industrialization pushes the type of rule into the tails. (p. 23-24).