Original credit to u/okteej over at r/askscience. I would have done a link to their question, but I couldn't figure out how.
As /u/Suedie correctly pointed out, this is usually a question for a linguist, but in the case of Egyptian phonology it is also a question for historians, because by historical accident the study of Egyptian language is deeply embedded in Egyptology. All that means is that specialists in Egyptian language also have to study (and take awful qualifying exams on) the seemingly-unrelated subjects: art, history, and archaeology as part of a PhD program, which makes them historians by any standard. So you've asked in the right place. You've also asked the right question, because this is exactly what my research is about.
Unfortunately, I'm too busy finishing up a talk for tomorrow's lunchtime colloquium to write out a full and direct answer to your question. But fortunately, I answered a very similar question on /r/ancientegypt some time ago, in which I coincidentally discussed the vocalization of Tutankhamun and Osiris. In short, Osiris is the Latinized version of the Greek version of Egyptian: ⲟⲩⲥⲓⲣⲉ ([ʔu'sirɛ]) and Tutankhamun is: ⲧⲱⲟⲩⲧ ⲟⲛϩ ⲁⲙⲟⲩⲛ ([tōʷt ɔnh amūn]) from something originally like: [tāwɪt anxā amāna].
I'll copy the relevant portion of that answer below:
So we now know that Egyptian is related to Coptic, and we know that it writes words phonetically. What's the problem then? Why can't we just read Egyptian like any other language? This is where it gets a little complicated. First, Egyptian isn't purely phonetic. Phonetic signs are almost always used in conjunction with ideographic signs (picture writing) to spell a word. For instance, the word "man" is written 𓊃𓏤𓀀 (zj) and the word "woman" 𓊃𓏏𓁐 (zt). The seated man and woman aren't phonetic, they tell you what sort of thing is spelled by the phonetic signs 𓊃 and 𓏏 (in this case, literally a man and a woman, but it's usually more abstract than this). Second, there are no vowels. The phonetic signs shown here are only the consonants. We know that 𓊃𓏏𓁐 zt was probably pronounced like "sit", because it might be the source of Egyptian Arabic ستّ ([sɪtt] = "woman"), and it is certainly the source of Coptic ⲥϩⲓⲙⲉ ([shimɛ] ← 𓊃𓏏𓁐𓈞𓏏𓁐 zt-ḥmt "woman-wife"). With 𓊃𓏤𓀀 it's even more complicated. There is a Coptic prefix ⲥⲁ- ([sa]), which derives from it, but we can't take that vowel sound at face value. In unbound, stressed position it was probably [si]. In sum, hieroglyphs don't record the vowels, but we know what the vowels probably were in these words because they show up in later Coptic words in a predictable way.
Ironically, most Egyptian names aren't arbitrary sequences of sounds, they are phrases in the language. Greek names can be used to decipher Egyptian, because they must be written phonetically, but true Egyptian names are made of normal words, which means that we can use Coptic to determine what they sounded like (approximately, more on this below). The two names you mentioned are great examples of this. Tutankhamun is 𓇋𓏠𓈖𓏏𓅱𓏏𓋹 (rearranged without the god's name first: 𓏏𓅱𓏏𓋹𓇋𓏠𓈖) twt-ꜥnḫ-jmn in Egyptian, which means "the living image of Amun" (i.e. he's an statue of the god Amun who actually walks around and talks to people: "The living Amun statue"). In Coptic, this is: ⲧⲱⲟⲩⲧ ⲟⲛϩ ⲁⲙⲟⲩⲛ ([tōʷt ɔnh amūn]), which was probably something like [tāwɪt anxā amāna] during his time. Nefertiti is 𓄤𓏏𓇍𓍘𓏭𓁐 nfrt-jj.tj, which means "the beautiful [lady] has come". In this case, every part of the name survives into Coptic (including the verbal suffix, which is quite rare). It would have become ⲛⲟϥⲣⲉ ⲉⲓⲧ ([nɔfrɛ it]) if it had survived that long, which probably would have been something like [nāfɪrat̚ iti] in her time.
Names are great for this sort of thing. Because they have to be transcribed accurately in other scripts, they preserve a lot of phonetic data. For instance, my username comes from a Late-Period Egyptian name: 𓊨𓁹𓀭𓈖𓆱𓐍𓏏𓏭𓀭 (wsjr-nḫty = "Osiris is strong") that gets transcribed as Ὀσαρναχθις ([osarnakʰtʰɪs]) in Greek, and would have been something like ⲟⲩⲥⲓⲣⲉ-ⲛⲁϣⲧⲉ ([usirɛ naʃtɛ]) in Coptic. I specifically chose this name becuase it has the rare quality of being written in every single Egyptian script plus Greek, but there are other names that preserve important pieces of information in the same way. Some New Kingdom names are even written in the cuneiform script in the Amarna Letters.
The reason for reconstructing an Ancient Egyptian antecedent that is different from Coptic is that the language changed considerably in the nearly 2000 years that separate the New Kingdom from Late Antiquity. A good comparison would be The Canterbury Tales in Middle English. If you listen to that recording with a modern translation, you'll see that it sounds different, but not completely different. It's clear that these languages are closely related, and after going through enough of the Canterbury Tales with a translation, you could easily take some modern English word that you've never seen in Middle English and project it backwards to create a hypothetical reconstruction of what its Middle English ancestor would have looked like. This is essentially what I've done to recreate the original pronunciation of Tutankhamun and Nefertiti. Using names and some other words attested in later languages, it is possible to reconstruct the sound changes that happened in Ancient Egyptian and create general rules for recreating things that are not fully attested later.
Further Reading: