I guess the transcription of the name *t3 wnnyt* is probably *touaeianaeine* (Crum Coptic Dictionary p.484b, perhaps citing this instance). It is interesting (to me at least) that a woman with so obviously an Egyptian name as Tamosis should be described as a Greek. I don't know these mummy label texts, but I imagine that the Egyptian is written in Demotic. The fact that it is written in Greek letters doesn't really surprise me. After all, the mummy label is only an identification tag and if, for example, the dead woman is being sent home, it was probably just regarded as a courtesy to put her name in Greek so that her possibly non-Egyptian reading relatives might be able to identify the body. Actually, I wonder how many people could read the scrawl generally known as Egyptian Demotic anway. Since I am now entering the realms of romantic fiction and possibly insulting Demotic fans, I will stop.
Minor evidence of bi-culturalism is the use of Greek for "external appearances" and Egyptian for "internal reality". Example:
In the Coptic letters from the 4th cent AD village of Kellis in the Dakhla Oasis there are instances of people with a Greek name and a Coptic one, e.g. someone might be named Theognostos in the address and Louishai in the body of the letter. It is not uncommon that someone called Psais in the address is later called by the Egyptian form Pshai in the body of the letter.