Personally, I think genetics does play a part in how tall you will eventually become. For example, I have two daughters, both with the same father. One is in the 80th percentile for height (4' 5" at 9 years old..hmmm...same as the British girls who are supposedly taller) and the other is about the 5th percentile for height (3' 11" at 8 years old). Since they live in the same household, received the same prenatal and preventative medical care and eat the same meals, according to the theory above, they should be roughly the same height when compared against their sibling at the same age. They are not. At age 8, my oldest daughter was 4' 2" tall.
I myself came from a family that did not lack for anything and we all had more than adequate health care and nutrition. My mother was 5' 6". I'm 5' 3".
While genetics plays a big role in determining your greatest possible height, whether or not you achieve that height does depend on good nutrition and health care, but to say or imply that stature is influenced soley by nutrition is false, IMO.

I am not familiar with European Newspapers and have no idea about the reputation of the paper from which this article orginates, but I always question statistics that do not include citations from researchers published in peer reviewed journals.

Statistics are very easy to generate, valid conclusions are another matter all together.

Who are the researchers? Where did they publish their findings?

I read the original article in The New Yorker that this article was based on and that is exactly what it said -- most size differences within populations are because of genetics, but average size differences between populations are the result of nutrition. Some people are genetically programmed to be shorter and some people are genetically programmed to be taller. However, that doesn't explain shifts and differences among the same population -- the Dutch are much taller on average than they were a generation ago and those of us in the United States are actually getting somewhat smaller than we were a generation ago (even adjusting for all the immigration).

I read years ago that a whole generation of british people were short, after ww2, food rationing continued for several years, supposedly the officials miscalculated the amounts necessary. Don't know if this is true or not.