This has big effects. In our timeline, the Shia Buyids in Iran and Iraq were overthrown by the Sunni Seljuks in the 11th century, and the Shia Fatimids in Egypt were overthrown by Saladin in the 12th century.
If Egypt remains Shia, the whole balance of power is altered. The Assassins based at Alamut were Shia Ismailis; their activities were partly a reaction against the collapse of Shia power at the time.
A Shia Egypt is going to exert a lot of pull on Syria as well, which it typically rules. It's not impossible to imagine a Shia crescent extending from Egypt through Syria into Iraq and Iran.
Would the Shia in Egypt be able to rule the Hijaz? If so, Islam today could see Shia as the majority form of the faith.
Later history becomes hard to fathom. The Safavids made Persia Shia partly to oppose the Sunni Ottomans. But would the Ottomans ever emerge in this scenario? What about North Africa? The Fatimids originally came from the Maghreb and that region was Shia for a time too.
It all depends partly on how successul this Shia Egypt is at projecting its power. It's certainly plausible that it never gets conquered by the Ottomans. In that case, the whole Arab world's history would likely be very different as the entire history of Ottoman neglect and decline followed by European colonialism likely never happens.
In that case the middle East as we know it today would simply not exist. In its place would be entirely different countries, likely with drastically changed political and economic circumstances. Who knows how it would all affect the rest of the world? We might see an Egyptian empire unheard of in modern history.