But that´s not true, the population of Latin American grew nearly 4 times from 1950, at the same time the USA grew only 2 times(Europe barely of 2/5). The birth rates of the 20th century removed any kind of imbalance and there is no point to rely on century old PODs to change anything.
How is what I said not true? Combine North and South America's populations, look how low they are compared to the Old World. Europe alone has 3/4 the amount, which is a historic low proportion to the Americas. Less than 1 out of 7 people globally live in the Western Hemisphere, despite the vast amount of land and food production potential.
So what if you had more people when the 20th century birth rates/decline in death rate take effect? That's very easily doable. For instance, if Argentina had the population density of the United States, then it would have over 94 million people. Take another instance, Canada--if it had the population density of Russia, it would have just over 84 million people. Obvious these are just raw math, and it would be highly improbable to get the population that high in those two countries, but these are illustrations showing how empty the Americas are, especially Latin America where in some parts the precolonial population levels were not reattained until the 20th century or later (1950s).
That's just countries, their subdivisions are further examples. Why couldn't the Falklands have even a fraction as many people as Iceland despite the similar climate? Why is Alaska so underpopulated compared to Scandinavia despite land favourable for agriculture? Speaking of the US, look at Montana and North Dakota, where in many counties, the population never returned to the levels of the first two decades of the 20th century despite the land being capable of supporting agriculture, just not profitably for market circumstance at the time. North Dakota's population was effectively static in the low-mid 600,000 range from 1920 until the oil boom there last decade. It's pretty clear that the Americas had and have far, far more potential for population size than they did, just historical circumstance effectively conspired to constrain it. In OTL, at least.
Then with the centuries old PODs, what if the Gulf Coast had settlement as intensive as that of, say, Virginia or the Carolinas? Yes, disease claimed many lives and stymied a lot of the growth there, but those are still people on the ground. Further, it opens the Upper Mississippi to settlement, where disease is no more of an issue than in Europe. All over a century earlier. That's gotta have some effect.
This is effectively a "free" boost to world population, allowing it to be much higher. I put "free" in quotes because these people need to eat (affecting land use for agriculture, which incidentally makes marginal land like much of the Dakotas/Montana much more economically sensible) as well as resource consumption, which is a much more difficult problem to solve. The immigration would have to be between 1500 - 1800 for the most part or so to avoid denting the population growth of Europe/Asia. Or, hell, put a huge dent in the population growth there, an Africa-sized one maybe even (probably infeasible, though), to really alter the balance between Western and Eastern Hemisphere proportionally.
From a POD of 1600 or so, I think 1.5 billion in the Americas is perfectly acheivable by 2016 without altering too many societal/historical trends (no hard natalist policies, still World Wars but no nuclear wars, etc.). Two billion is probably pushing it, though.