Prompted by a comment below I decided to "quick & dirty" double-check whether the Uygurs' East Asian ancestry was more Han or Japanese in affinity. My recollection was that it was more shifted toward sub-Siberian populations, such as Mongolians, who also are suggested to have affinities with Koreans and Japanese. My method was simple, I ran ADMIXTURE in supervised mode with a sample of 10 Han and 10 Japanese I randomly yanked out of the HGDP along with Uygurs, and assorted other groups. I left Uygurs, Chuvash, Turks, and Mongolians "free" to be linear combinations of the ancestry of other populations. The marker set was 290,000 (I pruned somewhat from 500,000+), and K = 11 (there were 11 populations which were part of the supervised run). It looks like the Uygurs have more of an affinity to the Japanese, or at least their East Asian ancestors did.
Of course, it could be that I sampled an unrepresentative class of 10 Han or Japanese. Much more likely in the former case, because the Han do exhibit some north-south structure. But I doubt it, I've seen similar results in other runs. A few other things to note is that the Uygurs have representations of the Pathan and Kannadi (South Indian) components. The last result is robust, and suggests the bare hint of "Ancestral South Indian" in the Uygur. I think this is evidence of gene flow over the mountains from Northwest India in historical times. Finally, there is the expected representation of European-like elements, but a substantial West Asian signal as well. The "West Eurasian" element which can not be attributed to later Indian & Iranian influence probably came with Indo-Europeans, and the results from the Uygur seem to support the supposition that many of the Indo-European groups were admixed between these two broad components.