I've always figured Arabia would sooner or later enter the greater Mediterranean cultural and political orbit anyway. And that Arab population increases meant there'd likely be some migrations. Given that the Romans and Sassanids used Arab mercenaries, proxies and foederati-I expect you may have seen some sort of repeat of the Germanic migrations. Though perhaps-the Byzantines would have held onto Egypt and Syria-I could see large Arab migration to these regions anyway, either through peaceful or unpeaceful means. The byzantines might hold political sway still-but there'd be a degree of Arabization, even if the Arabs were Romanized/Christianized/Hellenized/Persianized?
The Visigoths were perennially unstable-and sooner or later would have collapsed. And I could see the Franks, and yes Berber mercenaries and adventurers gaining clout or outright territory in Spain.
Geopolitically-the Sassanids were exhausted, and fought a civil war or two IIRC, before the Arab invasions came hard. I expect either that dynasty would have been overthrown or it would have reconstituted itself.
And thus the cycle of Roman-Persian wars would continue in some way-simply to the competing geopolitics and geographic location, it makes conflict inevitable. We see that with the Ottoman-Safavid conflicts as well.
Though an 8th round between the Romans and Sassanids specifically is questionable, if possible.
The Byzantines are going to lose control of Italy sooner or later. They might hold on longer due to not constantly defending Anatolia, and I could see a lasting Byzantine control in southern Italy-but everything north of Rome is simply untenable in the long run.
Other effects are harder to predict-Central Asia remains a meeting ground between different religions longer, and Christianity continues to grow in Persia, India was divided at this time, and would likely remain so.
Tang China probably reaches the limits of what it can control in Central Asia-internal rebellions will mark its end. Though without an Arab victory at Talas-the Chinese likely have the nominal allegiance of any polities in that region(the Tarim basin being that) for quite a while.
The Turks and other nomads never convert to Islam-this doesn't stop periodic nomad invasions into Europe, India, and the Middle East.
Beyond that-its impossible to say. Islam is just so important to world history, it was a sort of reset in some ways-changing just about everything, even beyond where it conquered. After a while-an alternate history writer dealing with this scenario can just plausibly make things up.
