There are a number of interesting assumptions in your original question that make this question a bit more difficult to answer. I'm going to go through your question step-by-step, to see if I can shed some light on this. In my answer, I'm going to focus mostly on the context in the Arabian Peninsula, as it is probably the most relevant for your question, and it's also the topic I know the most about. This is not to dissuade other people from giving their perspective on the role of monotheism in other parts of the Middle-East, however.

There are two ways of answering this question: firstly, we can look at what the historical material from the pre-Islamic period tells us. Secondly, we can look at what the Qur'ān and the Islamic tradition say. I'm going to do a bit of both.

As I understand it, Islam was created around the 6th century CE, shortly after the fall of the Roman Empire, but before the crusades and the mass spread of Christianity. Judaism was never a widespread religion.

So starting off with a bit of a nitpick: most people would say that Islam was founded by the Prophet Muhammad around the year 610. Traditionally, Islamic theology holds that Islam originated with Adam and that Muhammad was just restoring it, but that's not immediately relevant for our question right here.

What you consider the Roman Empire and whether it fell before Islam came into being is also a matter of subjectivity. Yes, the Roman Empire had been split in two several centuries before Muhammad's birth and the Western Roman Empire had fallen in 476 CE, but the Eastern Roman Empire – also known as Byzantium – was still very much alive and kicking. Or at least they had been before the extremely costly wars with Persia, which interestingly took place just before the coming of Islam. I'll get back to that in a second.

Thirdly, the concept that Judaism was "never a widespread religion" is also one that displays a certain amount of subjectivity. Yes, Judaism certainly never spread to the same degree that other religions (not just Christianity and Islam, but also Buddhism and Hinduism, for example) did, but Jewish communities had spread far beyond Israel, even before the final destruction of the Temple

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Presumably in the Middle East at the time most cultures or peoples wouldn't have been Judeo-Christian, or perhaps not even monotheistic.

So this is how we are going to get to the crux of the matter: presumably most cultures or peoples wouldn't have been Jewish or Christian, or even monotheistic. Leaving aside the entire debate on how to define monotheism exactly, there are a number of extremely important political and religious changes that took place in the two centuries preceding the founding of Islam.

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Judaism and Christianity

Outside of Israel, there are only two examples of Judaism having been instituted as an official religion: in the Khazar Khanate, which ruled over what is now Ukraine and Central and South Russia, as well as in the Himyarite kingdom in South Arabia. This is pretty relevant for your question: during the 3rd century CE, both Christianity and Judaism both arrived in South Arabia at around the same time. It seems that Judaism probably spread along with Jewish merchants, whereas Christianity was spread by missionaries (although there are no good contemporary sources directly confirming this).

We know about the spread of these two monotheist movements to South Arabia due to the considerable amount of Ancient South Arabian inscriptions from this period. Although the rulers and peoples of South Arabia had worshipped various deities, from this moment onwards they begin to worship a single deity. What's interesting about these early monotheist inscriptions is how non-specific they are, referring to God as "the Lord of Heaven and Earth" or "He, who created all things". These are certainly monotheistic, but not outwardly Jewish or Christian.

It is only during the 5th century, some two hundred years later, that Himyarite religious inscriptions take on an explicitly Jewish character. For example, we find references to synagogues as well as the usage of terms like "the people of Israel", and "the Lord of the Jews". The reasons for the Himyarites' conversion to Judaism are not entirely clear, but they they might have had something to do with the spread of Christianity to Ethiopia.

You see, Ethiopia has been a Christian country for a very, very long time. There is something of a dispute between Ethiopia and Armenia about which of the states accepted Christianity first, but it is clear that the Ethiopian ruler Ezana, who ruled from 320 – 360 CE converted to Christianity and made it the official religion. It is very possible that political conflicts between Ethiopia and the Himyarites played a role in the latter's decision to finally pick Judaism, but again, this is far from certain.

What is clear, however, is that religious conflict between the two nations finally led to the demise of the Himyarites. The alleged persecution of Christian missionaries in South Arabia caused an Ethiopian invasion in the year 521. In turn, this led to a South Arabian revolt, led by a local Himyarite nobleman, called Yūsuf Asʿar Yaʾṯar. According to near-contemporary Christian hagiographies, Yūsuf wiped out the entire Christian population of the South Arabian oasis town of Najrān in 524 CE. This is seemingly confirmed by a South Arabian inscription written at the same time. Either way, the Ethiopian response was swift, and South Arabian independence was ended permanently.

You may ask, at this point, whether the political struggles between these two nations affected the faith of your regular South Arabian or Ethiopian Joe Schmoe, but it is clear that both Judaism and Christianity were widespread.

The Islamic tradition

In this case, the Islamic tradition may be of some use. It is telling that Ibn Ishāq's biography of the Prophet Muhammad begins with telling us about the situation in Arabia before Muhammad's time, and tells us about how Judaism and Christianity had come to the Arabian Peninsula. The Islamic tradition also tells us about the mass execution of the Christians of Najrān, calling Yūsuf by the moniker dhū Nuwās, "he of the hair lock", and connects this Qur'ānic verses mentioning the Companions of the Pit ( ʾaṣḥāb al-ʾuḫdūd; Q85:4-8.) (although it must be noted it is not entirely clear this is what the Qur'ān refers to).

The Islamic tradition also tells us about the three Jewish tribes who lived in Yathrib, later Medina, known as the Banu Nadir, the Banu Qunayqa and the Banu Qurayza. These pieces of narratives indicate that both Judaism and Christianity were certainly a part of the day-to-day life on the Arabian Peninsula. And when we look at the Qur'ān itself, there are frequent references to both Christianity and Judaism, indicating that it came to being in a milieu that was familiar with both religions. Here it is worth noting, however, that the Qur'ān may have fundamentally misunderstood some of the theological underpinnings of Christianity: for example, it gives the impression that the Trinity consisted of God, Jesus, and Mary.

As a final note, it might be worth noting that despite the spread of monotheism to Arabia, there were certainly still polytheists among its population. This is most evident by the numerous references in the Qur'ān to širk, "association". According to the Qur'ān, the worst imaginable sin is ascribing partners to God, i.e. believing that there is more than one God. The Qur'ān itself also refers to three of female deities, al-Uzza, Manāt, and al-Lāt, whom we also encounter in some of the inscriptions of the area. So while the Qur'ān certainly came into existence in a milieu that was familiar with monotheism, traces of pre-monotheist Arabian religion also lingered.

tl;dr: both Judaism and Christianity were present on the Arabian Peninsula, and people were familiar with both

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Sources

Bowersock, G. W. 2012. Empires in Collision in Late Antiquity. Waltham, Mass: Brandeis University Press.

Crone, Patricia. 2010. “The Religion of the Qur’ānic Pagans: God and the Lesser Deities.” Arabica 57(2): 151–200.

Gajda, Iwona. 2009. Le royaume de Himyar à l’époque monothéiste: l’histoire de l’Arabie du Sud ancienne de la fin du IVe siècle de l’ère chrétienne jusqu’à l’avénement de l’islam. Paris: De Boccard [u.a.].

Ibn Isḥāq, Muḥammad. 2001. The Life of Muhammad: A Translation of Isḥāq’s Sīrat Rasūl Allāh. ed. Muḥammad ʿAbd al-Malik Ibn Hišām. Karachi ; New York: Oxford University Press.

Moberg, A. 1924. The Book of the Himyarites – Fragments of a Hitherto Unknown Syriac Work. Oxford: Oxford University Press.

Nebes, N. 2008. “Die Märtyrer von Nagrān Und Das Ende Der Ḥimyar. Zur Politischen Geschichte Südarabiens Im Frühen Sechsten Jahrhundert.” Aethiopica 11: 7–40.