They didn't go anywhere. They just changed (and assimilated).

Alexander the Great conquered Egypt (hence the city of Alexandria), during his was against the Archaemenid Empire (of which Egypt was a tributary). After his death, his empire was divided up, and his general and childhood friend Ptolemy I Soter took over Egypt. He was a Macedonian-Greek.

So, Egypt was now ruled by a culturally Greek dynasty. To gain the support of the locals, they followed many Egyptian customs, but they were still a Hellenizing influence on the Egyptians, and most officials were Greek.

Over time, Ptolomaic Egypt declined in power, and so allied with the rising power of Rome. This alliance became more and more one-sided, with Egypt supplying vast amounts of food to Rome. Thus, when Egypt was plunged into civil war by a succession crisis, threatening Rome's food supply, Julius Caesar intervened, putting Cleopatra on a throne. After his death, she allied with Mark Anthony. Anthony became very unpopular in Rome, and his rival Octavian attacked and defeated Anthony's fleet, and then invaded Egypt. Anthony and Cleopatra killed themselves, and afterwards Caesarion was executed. Egypt was formally made a province of the Roman Empire.

Under the Romans, the Egyptians were allowed to keep their own culture. Greek was still used as the main administrative language, and there was some syncreticisation between Roman and Egyptian Gods (continuing a process that had started under the Ptolemies). For example, Hermanubis: a mix of Hermes and Anubis. https://en.wikipedia.org/wiki/Hermanubis#/media/File:Statue_of_the_god_Anubis.jpg The Roman Imperial Cult was also slowly introduced.

So, over the next couple of hundred years, the already somewhat Hellenised Eygpt continued to exist in the Greek-speaking world, as part of the Roman Empire.

Of course, Egypt was close to the Roman province of Palestine, the origin of the new and growing religion of Christianity. Alexandria became one of the core areas of Christian activity, and the Egyptian desert was popular with the "desert fathers", hermits who evolved into the first Christian monks.

Like the rest of Rome, Egypt had a small but growing population of Christians, which exploded in the 4th and 5th centuries. There were of course still pagans around, but Egypt was much more successfully Christianised than western provinces.

Over the next couple of centuries, however, as the power of Rome declined (the Western half essentially collapsing), the Egyptians became more and more resentful of the Empire. Egypt was one of the richest and most fertile parts of the Empire, yet they were paying taxes and food to the Empire. This was expressed in arguments about what might seem to us obscure theological points. Greek theologians, supported by the Emperor in Constantinople, insisted that Jesus Christ was both 100% human, and 100% divine, at the same time. Most Egyptian theologians, however, argued that Jesus was simultaneously huamn and divine, mixed together. Which... is not really very different, to be honest, but the fallout was primarily about political tensions. In the early 7th century, Persia briefly conquered Egypt, allowing local versions of Christianity to thrive without persecution from the Imperial-controlled Church. While Rome did reconquer Egypt, the issues got worse.

In addition, the war with Persia had severely weakened the (by now purely Eastern) Roman Empire's army.

Thus, when Caliph Umar sent an Arab army into Egypt, Rome had little local support, and a lack of military support. After a year, Egypt fell to the Arabs.

Under the following Ummayed dynasty, Egypt was ruled by Muslim Arabs, but most peasants and villagers were still Christian, and were largely allowed to continue their religion unmolested. From 1000AD onwards, however, Egyptians converted at a much higher rate, and by the late middle ages only 10-20% of Egyptians were still Christian. Today it stands at about 10%.

Under the Abbasids, however, Egypt didn't just Islamise, but Arabised. It was part of an Arab world, ruled by Arab speakers. Arabs emigrated in, traded, travelled through. Over time, Egyptians largely stopped speaking Egyptian (which now survives mainly as the language of the Coptic Church) and started speaking Arabic. They married into Arab families and networks.

That leads to today's Egypt, where Egypt is an Arab-speaking, 90% Muslim country.