The interplay between nutrition, stature, and status is pretty well understood, though the most abundant empirical data comes from the "modern" era when consistent records of height measured consistently came into practice. Cemetery and archaeological studies also have the confounding factor of having to determine status using things like grave goods and location within the burial site, if there are not obvious social distinctions in burial practice (i.e., tomb vs no-tomb). Further confounding factors are generally low sample numbers of available skeletal materials, which themselves may be fragmentary which can lead to various alternative methods of estimating stature. Generally, historical stature estimates work at the population level, not the class level, for these reasons.

Nevertheless, there have been studies which evaluated stature differences as related to status differences. The results, however, may underwhelm you. Vercellotti et al. (2011), looking at northern Italian cemetery dating from the 8-13th C. found the stature difference between those they classified as high-status males and those they classified as low-status males to be about 3cm (171.59 vs. 168.24). Larsen in his (1997) Bioarchaeology references a few studies ranging from prehistoric American Southwest to Bronze Age Greece which similarly have shown a difference in stature attributed to social status, with the Greek sample showing a difference of 6cm between high and low status groups. Haviland (1967) similarly estimated that an early Classic Maya site may have had differences between elite and non-elite males of 7cm, though he was working with some very small sample sizes. This larger numbers are best seen as outliers though. Returning to Medieval Europe, Schweich & Knusel (2003) compared remains from a 12-18th C. urban hospital and leprosarium to remains from a 12-14th priory cemetery composed of "mid to high status" burials. While the overall average status of the priory remains were taller, they were not significantly so (~1cm).

The range of no more than a couple of centimeters between population sub-groups can be seen in more modern examinations in status/stature disparity. Steckel (1995) notes that the stature difference between a sample of Ohioan "professionals" and "laborers" was only about 2cm (175.5 vs. 173.3). This narrowing of difference could be attributed narrowing of social disparity compared to Bronze Age Greeks or Classic Maya. For instance, Steckel and Rose's (2002) The Backbone of History: Health and Nutrition in the Western Hemisphere compared skeletal remains of several 19th century groups, including enslaved and free blacks, at various sites. On the 2 plantation sites, average adult male height was 168cm and 164cm, respectively. A church cemetery of free blacks in Philadelphia, in contrast, had an average adult male height of 171cm, which was in parity with a group of middle class Canadian whites. For further contrast, a group of Charleston elites adult male height averaged out at 175cm, which was shorter than a (tiny) sample of free blacks from rural Arkanasas (177cm). The problem with all these numbers though, is the perilously small sample sizes (sometimes single digits) leading to inevitable skewing of the data.

So, yes, we do consistently see elites in societies trending towards greater stature compared to the hoi polloi, but the nobility were not Eloi towering over misshapen Morlocks. We generally see disparities of a few centimeters. While the stature disparity does appear to increase in societies with greater status disparities, we also have to take into account the very small samples sizes and the problems of comparison. To really determine if the elites in society were on average taller than the non-elites, you need to have contemporaneous skeletal remains from the same society; from the same area within that society; with clear indicators of the separate statuses; and then compare equivalent groups (e.g., adult males). Add in the fact of fragmentary remains with small numbers of individuals and the extreme variation we see from slaves in South Carolina to freemen in Arkansas starts to make more sense as a statistical problem rather than a biological or nutritional one. Height, within certain limits, is also a very plastic attribute which even at the general population level shows ups and downs in historical populations. The priory cemetery remains I cited earlier? Just few centuries earlier, the average height of the parish in that location had been several centimeters taller than the later groups. Again, we can talk of general trends in population stature and even intragroup stature affected by social disparities, but we have to acknowledge that we are talking about a variable trait with multifactorial causation for which our data is not exactly the best.

Related to this, Joseph Heath mentions in Filthy Lucre that the expression "look down upon someone" comes from the fact that English nobles and wealthier people tended to be taller than commoners, so they literally looked down on them. Is that correct?

Heath, Joseph. Filthy Lucre: Economics for people who hate capitalism. HarperCollins, 2009.

I can't answer directly to nobels per se, but height differences between groups have been used as a measure of inequality in historical research. One article that comes to mind is the chapter on wage inequality in van Zanden et al 2014, How was life? Global well being since 1820. The measure he uses is the gini coefficient, but if you read into the data section you'll see that some of the estimates are based on height data instead of wage data, but that the height differences are expected to accurately reflect the differences in wage and income. However, heights are expected to reflect more than just caloric intake; they also reflect disease environment and other living conditions. Heights were a very trendy measure of historical living standards in the 80s and 90s, though they aren't so trendy now.

If this comment is appropriate I will link the chapter and fill more examples when I get home.

Edit: you can connect this to the herotability of wealth historically, to get a better idea of generational effects. Greg Clark's "the son also rises" traces wealth herotability. I'll remind myself of the content and update you, but it's a fun paper to read anyway because it makes people really angry.

In agrarians times, would nobles have access to better nutrition?

I don't have any link other than stories I heard from my grandfather. Japanese officers were significantly taller than the grunts in the Pacific War. They were obviously from a higher caste, and had better nutrition and caloric intake.