I don't know if it's the best series of data to use for this. I would have used a much longer series if possible because to be honest the past ten years have not seen a dramatic change in sugar consumption, but I guess their diabetes prevalence dataset only spanned that long. But that brings up the question of the lag for developing diabetes itself, it's not like 50 extra calories. I also couldn't find if they threw out countries with weird or obviously low quality data. Israel's data, for example, is a bit strange. It goes from around 500 kcal/capita/day sugar to 230. I find it hard to believe that consumption habits changed so drastically there.
They also do not discuss much about why they, as non-economists, chose to use econometric rather than epidemiology or biostat methods. I also hate that their final graph is totally unlabeled and there is no option to download a format that would allow one to see which dots correspond to which country.
It would be wise for some of the alarmists on this thread to read the paper and this sentence in particular " Hence, any of the findings we observe here are meant to be exploratory in nature, helping us to detect broad population patterns that deserve further testing through prospective longitudinal cohort studies in international settings, which are only now coming underway."
I looked all over FAOSTAT but couldn't find an explanation. http://faostat3.fao.org/home/index.html#METADATA