https://en.wikipedia.org/wiki/British_Agricultural_Revolutio... is the primary cause of this change. Until food productivity started growing dramatically, populations were generally kept in check by the threat of famine. And famine was more a problem for peasants than nobles.
The change happened earlier in the Americas for the simple reason that due to recent settlement, population density did not yet match what the land could sustain.
Certainly these improvements led to overall population growth, which accelerated over a few centuries. But the pattern that richer people had more kids than poor is a different thing. I believe this persisted until Victorian times in England. The US frontier was a gigantic anomaly in other ways, and could be in this way too, I don't think I've seen data.
The change happened earlier in the Americas for the simple reason that due to recent settlement, population density did not yet match what the land could sustain.