There are three separate threads to pull, here:

1. Philological reconstruction of the ancient Egyptian language from its descendants, of which Coptic is the most recent. Coptic died out as a living language in the 17th century or so, but its speakers left a sizable corpus of written documents in a Greek-derived alphabet that preserved pronunciation about as well as any other contemporary writing system. There are modern efforts to revive spoken Coptic, but I don't know offhand how much progress they've made.

2. Decipherment of Egyptian hieroglyphics, kickstarted by the discovery of the famous Rosetta Stone in 1799 and truly unlocked by the research of Jean-François Champollion in the early 19th century. It's a very complex writing system, but it's generally pretty well understood today. Interpreting vowels is generally pretty tricky, which is why you'll often see ancient Egyptian texts transliterated with consonants only: e.g., the name Nefertiti as it appeared in hieroglyphics would be "officially" rendered as Nfr.t-jy.tj.

3. Correlation with transliterations of Egyptian words and names by other language communities, especially the Greeks, as you point out.

Ultimately, our understanding of ancient Egyptian pronunciation is a reconstruction, not an observational analysis. It's a good reconstruction, as such things go, based on 200+ years of scholarship and analysis, but one should never lose sight of the fact that it's provisional and subject to adjustment based on potential future discoveries. The usual historical linguistics caveats apply, too: There is no single unified ancient Egyptian language that was spoken in the same way everywhere through time, it was a living language just like any other with contemporary dialect variation and many changes in phonology, grammar and syntax over a very long period of time (a whopping 3000+ years).

General references:

Antonio Loprieno, Ancient Egyptian: A Linguistic Introduction, Cambridge, 1995.
James P. Allen, The Ancient Egyptian Language: An Historical Study, Cambridge, 2013.

Adding to /u/suddenly_mozzarella's answer, I answered a question before about how we know what Roman Latin sounded like, which applies to your question as well in terms of the historical reconstruction point (mozzarella's number 3 + Coptic)

That answer is here.