What if Islam never exists and Christian Arabs conquer all that land? Is that permitted under these requirements? If there is no religious barrier, Arab civilization will be much more attractive to Europeans.
Well... In my opinion language is a
tool. A useful tool to swap ideas and other stuff.
In ancient times, in Mediterranean area, there were two "international" languages: Aramaic (since Persian Empire - VI BC, and Greek (since Alexander - IV BC). After Alexander, Greek displaced Aramaic language as international one, but in the Middle East there were areas (Syria, palestine, Mesopotamia), where Aramaic remained as "regional"/"second international" language.
When the Romans conquered the Mediterranean area, and a New Persian (Sassanid) Empire was established, Greek remained an international language (educated Romans were bilingual, it was a language of scholarship and diplomacy).
Aramaic was used again by the Eastern Churches to popularize Christianity in the Middle east, It was always used as a "second" international language there anyway. That made the role of this language more important, the use of greek was declining, the Aramaic-speaking Christian scholars were translating Greek books into Aramaic.
On the west, the economic crisis of Rome and further barbarian conquests made Greek out of use there, leaving Latin as
lingua franca there. There was not enough people speaking greek to keep it as useful tool. Similar thing happened as in the East - the Bible was translated into Latin.
So - at the eve of Arab conquest there are 3 regional languages to compete as future lingua franca.
In a TL with non-Muslim Arabs conquering the Middle East, they enter on Aramaic language territory - similar to Arabic, but already established as scholar and church language. As not many conquerors were able to write, it would've been easier for them to use an existing tool. So probably the second generation of non-Muslim (Christian?) Arab conquerors would probably learn to read/write and then speak Aramaic. Probably some of them would learn a few words of Greek as well.
If the conquest goes as OTL, we have TL with Aramaic-speaking world in most of Mediterranean. And if Arab/Frankish and Arab/Western Christianity connections are strong it
might give some impulse for Aramaic as
lingua franca. Not Arabic, I'm afraid.
To use this language as a
tool, we need stronger connections (cultural, trade) between Aramaic Arabs and Latin speaking people than OTL. With both groups as a Christians, it could've been stronger than OTL, but - Aramaic-speaking Arabs in this TL are members of Eastern churches, so the gap with the west still exists.
And with those connections there is still another tool to use - Greek. Aramaic scholars knew it well, and at least some West-European scholars knew it, or may use Byzantines from Italy as translators...
I'm sorry, but IMHO there's no chance for widely-spoken Arabic language without Islam. Aramaic might have a chance in this TL.