What is the relationship between 尉-shaped characters and the pronuncation yu?
The two syllables are related.「尉」(Baxter-Sagart OC: /*ʔut-s/) has two Mandarin descendants, which are approximately yu and wei.

Several OC syllables with initial /*ʔu-/ basically evolved into Mandarin w and Mandarin yu; you can see the same phenomenon with the phonetic component「」:


溫 wēn
慍 yùn