As the title says, what if the Islamic invasion began while Byzantium and Persia were still at war?
The obvious problem with this is that IOTL the war ended before Muhammad had consolidated control of Arabia. But it was a relatively narrow thing- the war ended in 628, Muhammad conquered Mecca in 630, and by 635 the Arab forces were invading Persian and Byzantine territory.
So we have two options:
1)Delay the ending of the B-S war
It's unclear to me whether the B-S war could plausibly have dragged on another 7 years- but perhaps it doesn't need to last quite that long. The Arabs may have begun their invasion earlier if the B-S war was ongoing, since attacking the Sassanian's exposed Southern flank while most of their forces are still tied up with the Byzantines some distance away is obviously a more promising prospect then attacking both empires simultaneously.
2)Delay the beginning of the B-S war
Perhaps if Maurice's assassination had been avoided/delayed?... the simplest and laziest POD might be to say that he get's assasinated in ~610. Failing that, it's still a matter of when, not if, the the next B-S war begins.
This also leaves the question of what effect a delayed B-S war would have one Muhammad's rise. Muhammad's first revelation supposedly took place in 610, 8 years after the war begun IOTL, and the trade-centered Hejaz would no doubt be deeply effected by the delayed war. But for conveniences sake I'm going to assume that Muhammads prophethood and formation of the Caliphate would follow a broadly similar path to OTL.
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Under either 1) or 2), I'm going to assume that the Islamic forces would cut the Sassanid's exposed Southern flank like a hot knife through butter. Given that they conquered all of Persia and much of Byzantium IOTL despite being at war with both simultaneously, it's hard to imagine them failing against Persia alone while most of the Persian forces are tied up with the Byzantines.
How things go from here depends greatly on how Khosrow responds to this catastrophic black swan(assuming he gets the chance to respond before being overthrown in a mutiny as he was IOTL). He could seek peace terms with the Byzantine emperor, so freeing him to march against the Arab forces. Or, he could underestimate the Arab threat, send a portion of his armies to suppress them while continuing to focus on the Byzantines, only to see his position collapse completely as the Arab armies sever the routes between Persia and their forces in the West.
As for the Byzantines... well, IIRC they initially perceived the Arabs as heretical "Christian" enemies IOTL. In this scenario however the Arabs are their salvation(or at least seem to be), so heretical Christian allies is likely to be be the immediate perception. Once the Persians have retreated, and it becomes apparent that the Arabs have no intention of returning Syria and Egypt to Byzantine governance...
I'm now realizing that this scenario is much less interesting then I'd initially supposed, because this seems to be bringing us to a similar endpoint as IOTL. The Persians likely still swallowed whole by the Caliphate, conflict between the Caliphate and Byzantium with the former failing to breach Constantinople...