This is all mostly correct, but a couple of minor points might be added just to be myopic
There is a triple transliteration going on.
Hebrew → Greek → Latin → English.
Each time a transliteration is made, different letters are used to represent the sounds of the earlier language, and that languages particular ability to speak changes the sound of the word slightly.
In Aramaic/Hebrew the ‘h’ sound was pronounced.
It would have been, hAlal Yahweh! – or out of respect hAlal Yah!
(But again this is just a transliteration, you have to guess at the sound).
The Y sound oftentimes is transliterated as J. (There was no exact J sound in Greek or Latin, they used the dual vowel ‘ie’ to make a sound close to J. e.g. Iesus Yesus Jesus are all pretty close in the way they sound.)
I do not know much about Hebrew/Aramaic, but many people who seem to know argue exactly which sound it was. (No 2000 year old tape recorders to end the debate!)
When the word went from Hebrew/Aramaic to Greek, the ‘h’ sound went with it to become
αλλελυια. This word is about as close as the Greek come to pronouncing the hebrew/aramaic. You will notice, if you look closely, that one letter appears to be missing -- instead a is present. That is called a rough breathing which makes the ‘h’ sound (it is a pronounced ‘h’ in the Greek).
Many Greek texts do not have the rough breathing – especially in the new testament. I am not sure why, but it is not uncommon for easily accessable texts to drop all breathing marks.
Latin people often borrowed words from the Greek, and when they did they almost invariably dropped the ‘h’ anyway. So the word became shortened to alleluia.
When Jerome came along, he checked both the Greek and the Hebrew (he knew both, but he knew Hebrew better). Because it was already a longstanding custom in Rome to pronounce the word from the Greek, which was a liturgy alive at the time as well as Latin, Jerome chose to leave the word as alleluia rather than retransliterate it from the hebrew. In other words, it was tradition which chose the sound we have.
So the word stayed in this form, the Latin one, for a very long time. Until it was again transliterated into the english from the Latin – as the well beloved alleluia. (Considering we have a roman alphabet, you might even argue it was just copied

)
It is funny how in English and Latin, the ‘h’ sound is still attached to the word. In hebrew the h preceeded the a sound, in english we reversed it! the ‘a’ used is pronounced like ‘ahhhh’.
as for the ‘j’ in hallelujah, I think it obvious that many protestants would rather have nothing to do with tradition.
Hope that helps.