Ah my history knowledge is mainly from Chinese language sources, so I may not know too much about English names.
More Chinese words.

Listed are some related words with their pronounciation in modern day mandarin Chinese.
The words (1) Hui-ren (Hui people), Hui-jiao (Hui religion) do come from Hui-he. Hui is the first character and abbreviation for Hui-he. In modern day Chinese language, "Hui" generally refers to Chinese non-Han muslims, but in the Tang dynasty, it meant specifically the Hui-he tribe. Hui-he is renamed to (3) Wei-wu-er in the Yuan dynasty, and used in Ming and Qing official documents. I just looked up the "Twenty-Four History", and in Yuan-Shi (History of Yuan) it does say Wei-wu-er and Hui-he were the same people. (4) is used in modern day Chinese. They probably changed it because (3) and (4) reads very similar, and (3) literally means "afraid of my son" in Han language.
Today we use (4) to address them because the word "Hui" has a much boarder meaning now.
But notice that these are modern day
mandarin pronounciations. The spoken dialect used by the emperor and the court officials changes from dynasty to dynasty. The Tang dialect is more similar to modern day
Hakka dialect than to mandarin. (2) was the official Tang translation of that tribe's name, but it could sound a bit different back then. (3) and (4) pronounce more or less the same.
So, all of (2), (3) and (4) could have pronounced pretty much similar to Uighur or Uyghur,
at their respective times.