A few thoughts
- What we think of as "Byzantium" may never emerge, or, if it does, it'll be in a considerably altered form. Without the tremendous religious and political power of Muhammad and his early successors to launch the Arabs across the Mediterranean, the Late Roman state will continue onwards for the foreseeable future. I would think that more attempts would be made to enforce the Monothelite doctrine, possibly at the expense of Italy. Heraclius, of course, is a triumphant hero, the greatest Emperor since Constantine, rather than OTL's rather sad and pathetic character.
- Traditional Iranian culture is probably already on the decline. In the absence of Islam, the House of Sasan may or may not be able to consolidate power from the terrible 630s. If it does, I think within a generation or two Nestorianism will be adopted by the dynasty. This may very lead to
Eranshahr fracturing between the Christians of Mesopotamia, and the determined old Zoroastrians of Parthia and Media.
- The Jews will hang on, no matter what predictions of doom and gloom are made for them, though in a world with a Christian Iran as well as Rome, it's going to be difficult. They'll endure sporadic persecution, but largely be left alone, as happened in the fifth and sixth centuries OTL. I can't really see any major reason for this to change. Sooner or later, they may get a Khazar-style lucky break, and convert a ruler somewhere or other- perhaps the Turks could be a candidate for this.
- The Arabs themselves will continue to Christianise, as they had been doing in the sixth century, though they'll adopt plenty of elements of their old faith, such as
Ka'abas. Odd Christian sects will probably emerge amongst the Arabs, and conversion by some to Judaism is not impossible. Demographic exhaustion in Rome and Iran will allow the Arabs to take advantage and raid Syria-Palestine and Mesopotamia, but it's unlikely they'll be able to hold onto all of this.
- With the Empire having even less desire than IOTL for conciliation, the Roman Papacy will find itself between a rock and a hard place, defended against Arian Lombards by an heretical Heraclian dynasty that's most unlikely to tolerate open religious dissent. Exarchs of Ravenna are likely to be reliable Easterners, and will make sure the Papacy doesn't step too far out of line. There may well be revolts in Africa and Italy, either seeking to overthrow the Heraclian line or (less likely) to re-establish an "Orthodox" Western Empire.
- The Visigoths will continue in Spain for the foreseeable future and may, if they are capable, start intervening in Africa and Italy for the cause of "Orthodoxy"- the same goes for the Franks. This could potentially be interesting for the Roman Empire to deal with.
- Then, there are wild cards. The Avars could recover their strength, one of the Caucasian kingdoms could get an unexpected Arab-style rise, a people could convert to Judaism somewhere that OTL would find bizarre.
The end of antiquity really is such a great time, isn't it?
