So apart from the Byzantines who were invaded, do we have any commentaries or mentions from people in the Lombard kingdom, the Frankish realms, or the Anglo-Saxon kingdoms? Were, for example, contemporary non-Greek religious writers like Bede aware of the Islamic expansion?
How did Franks, Goths or Anglo-Saxons reacted to the Arabo-Islamic conquests until the mid-VIIIth century? They barely did.
Latin sources are essentially silent on this matter, with the exceptional monk or clergyman actually coming or went to newly conquered part. One such rare testimony is the account of Arculf, a bishop from Gaul, made of his pilgrimage in the Holy land in the late VIIth century and written down by Adamnan, one of the most learned monks of the British Isles in "About Holy Places", De Locis Sanctis; and Bede's own description of it.
Arculf describes there the new mosques, mentions Saraceni, and Muhawiya by passing, but barely gives us information about them and rather focuses to what was important to him, the description of Holy Places and relics. Arculf doesn't really seem to understand the religious specificity of Arabs, and have the Caliph making a rather Christian-sounding speech and prayer when judging on a dispute between Jews and Christians, maybe considering Saracens rather as "unbelieving" heretics with their own "churches". Bede however rather than considering them as heretics, make them star-worshippers.When going to Constantinople in the way back, the conflict between Arabs and Romans doesn't seem to interest him nearly as much as the churches and saints of the city, and he makes no mention of them.
This account is essentially the only Latin source mentioning the Arab conquests in the VIIth and early VIIIth century; with some mentions in the Liber Pontificalis : the Syrian conquests themselves aren't accounted for (while Rome had a privileged connection with Syrian monasteries); but an expedition in Sicily is (mentioned with the assassination of the emperor, years after it took place), the brief reconquest of Africa, or the peace treaty of Justinian II. Barring that, no mentions of Arabs in this source before 715, meaning 30 years of silence on the question.
What we can learn there is that Latin Christians were aware of the Arab conquest on Romans, as they don't feel necessary to explain who Saracens were. But it appears they didn't cared much about it either, seemingly considering them as heretics in a region where theological pitfalls and heterodox teaching are legion already, and as long as Latin Europe isn't concerned, a non-event.
The growing differentiation between the western kingdoms and the Eastern Roman Empire was already ongoing by the VIth century, even if it's hard to attribute it to a single main cause : impopularity of Imperials due to the Gothic Wars and christology theological conflicts, decline of Mediterranean trade (and therefore subsides and prestigious gifts cementing Byzantine diplomacy since the late Vth century), and general re-polarisation of Latin West in Central and Northern Europe since the mid-to-late VIth.
While Goths, due to the power vacuum after the fall Carthage, satellited the remnants of the exarchate in westernmost Africa (notably Septem), it's not really clear how they perceived the newcomers, except as raiders (especially with Roman refugees going to Spain) and the first clear mentions appears only after the conquest, in the Chronicle of 754 made by refugees in Cantabrian highlands.
Even the conquest of Spain in 711-715 didn't immediately provoked a surge of attention, and it's really from the next decades, with the main raids and counter-raids in Gaul in the 720's and 730's, the migration of Hispanii, Spanish refugees in Carolingian Francia; that Latins source are more systematically interested.
That Eudon of Aquitaine was ready to ally himself with the amir of Cerdanya to secure the pyrenean passes against new raids, could well point that, in spite of Carolingian self-publicity (and Charlemagne didn't that hesitated allying with pro-Abassids in Spains) could point that Franks and Aquitains didn't saw Arabo-Berbers as an existential threat (arguably, Christians lords in northern Spain were of a rather different opinion).
Even there, they tend to be brief and for most of the period, and mostly about conflicts and when Alcuin credits Christians of Spain of having reconquered the equivalent of a good third of the peninsula in 790, one can cast doubt on the quality of the information the Carolingian court received.
- De Locis Sanctis; Adamnan
- De Locis Sanctis; Bede
- Les Carolingiens et al-Andalus (VIIIè-IXè siècles); Philippe Sénac
- Visigothic Spain 409-711, Roger Collins