In general, one would hope to find the depository that holds the holotype as a part of the formal description article published. However, with many, especially older (1800's or earlier) descriptions, this isn't necessarily the case. Often major collections, such as Kew (UK) or the National Academy of Sciences (USA) will hold collections from important researchers, but this is not always the case. Nowadays, you often have to do a bit of searching to find this information. I've put a description of my process below.
The Journal article in question is:
- Nuttall, T. 1848. J. Acad. Nat. Sci. Philadelphia ser. 2, 1(2): 179.
The link provided on the Wikispecies page doesn't take you to the article directly, only to a page for the journal, which links to series 1, volume 1.
I did a bit of link sleuthing and was able to link from the wikispecies page, through the International Plant Names Index (click on the top listing to take you to this page, where you click on nomenclatural link to this page, click on publication, which takes you to this page, where you need to click read this publication under heading BHL). All this finally takes you to the Biodiversity Heritage Library listing for the journal, where you need to access the right volume there (Volume: ser.2:v.1 (1847-1850)), listed as 1847, but covering 1847-1850. You want to look at page 179 under the heading Quercus for the description.
The article doesn't provide any information on the location of the holotype, so a bit more, very easy, sleuthing and you will find that Thomas Nuttall, the descriptor, worked at the Academy of Natural Sciences Philadelphia (now Academy of Natural Sciences of Drexel University), and unsurprisingly published in their periodical. This makes it highly likely he deposited his material with them. From a quick search of their site, it seems they do indeed hold some 4,000 samples from Nuttall.
They also hold some records from W. Gambel, in their Historical collections. Nuttall's article in the Journal of the National Academy of Sciences, Philadelphia was a description of samples collected by Gambel, and Quercus gambelii is named for him, so they should(!) be with either Nuttall's or Gambel's collections.
So, long story short: The specimen is likely to be at Drexel University in Philadelphia.
I say "should(!)" because things get lost, sent off to other places and not returned etc. Nuttall also worked in the UK from 1842 onwards until his death, as he inherited property from an uncle, under whose will, Nuttall had to spend at least 9 months of each year in England. I don't know enough about the chap to say whether he still continued to work in Philadelphia, but he certainly continued to publish there, as evidenced by the article for the species in question. This could mean that some of his specimens are in the UK. There doesn't seem to be at Kew, at least not digitized, but there are quite a few other sites they could be in the UK.