When historians say that 'Islam engaged with these debates', they do not mean that we have evidence of these debates, but debates in a more metaphorical sense, as in: the things Islam (or proto-Islam, or the Believers movement etc.) talked about were the things other contemporary sources were talking about. Issues such as the apocalypse, holy war, and the nature of God were the most widely-debated topics of the sixth and seventh centuries. Early Muslims/Believers must have dealt with these ideas and probably did talk to other Christians, Jews, and the mysterious hanifs about them, since we can find clues in the Qur'an and the only other contemporary document from this time, the Constitution of Medina'. Take for example the sura dealing with the war between the Romans and the Persians (603-628):

The Romans have been defeated in a land nearby. Yet after being defeated they will prevail within a few years - unto God belongs the affair, before and after, and on that day the believers shall rejoice in God’s Help.

We have no source telling us how Muhammad learned about this conflict, yet he evidently knew about it. The same applies to other topics and Peter Sarris provides a succint summary in his Empires of Faith (2011):

The religion of Muhammad as it emerges from the earliest sources was a faith deeply engaged with wider debates and concerns that were coming to dominate the religious life of the Mediterranean world into which Arabia, from the fringes of Syria to the Yemeni littoral, was increasingly being drawn. The verses of the Qur’an reveal a preoccupation with the imminence of the apocalypse, and a determination to counter those who denied the existence of the afterlife, such as we have already encountered in the writings of Pope Gregory the Great. Like Gregory’s Book of Pastoral Care (or the Spiritual Meadow of the seventh-century Palestinian monk John Moschus, who wrote under papal patronage), its verses offer an intensified but pragmatic piety whereby the core asceticism of the ‘Holy Man’ was available to all. The vision of paradise contained in the work, and the rituals associated with prayer that we encounter in early Islam, moreover, bear a striking resemblance to those found in the early Christian Syriac ascetic tradition. The respect in the Qur’an for the Virgin Mary, but its denial of Jesus’ divinity and crucifixion, chime closely with shades of contemporary Christianity, both ‘orthodox’ and ‘heterodox’. Indeed, it is striking that a Christian heresy is precisely what the eighth-century Syrian theologian John of Damascus regarded Islam to be. What Muhammad’s austere monotheism cut through were the complexities and controversies of Christian Trinitarian and Christological doctrine. In that sense it was itself a fulsome response to contemporary Christian debate.

[...]

That Muhammad should have preached a doctrine of martyrdom and holy war at precisely the time that these selfsame concepts were featuring so prominently in the official pronouncements of the East Roman state is highly suggestive, and may indicate that Heraclian propaganda was percolating into the Arabian peninsula. The region around Mecca appears to have been one of the few parts of Arabia that had not come under direct Persian control by the late sixth and early seventh centuries, and thus it may have been something of a bastion of pro-Roman sentiment. Certainly, sympathy for Heraclius’ cause is strongly indicated by the Qur’anic verse on ‘the Greeks’ (al-Rum), possibly recording the Persian defeat of Roman forces near Bostra in 613.

Others take it further, one study for example noting the influence of Syriac literature on the Qur'an,^1 but to discuss this topic in more detail we have to first discuss what we mean by 'Islam'. The answer: no-one can be sure yet. We simply lack the sources to say anything for certain and scholars working on this topic range from the late Patricia Crone, who argued forcefully for Jewish/Jewish-Christian influence on Muhammad and his beliefs,^2 to others who are more or less content to follow the traditional sources. Tom Holland's documentary is great in that it managed to bring to the public's attention a particularly (in)famous strand of thinking about Islam, but it's not the most convincing one - Crone's idea about the position of Mecca is particularly weak in my opinion. A more moderately revisionist, and perhaps more persuasive as a result, argument is laid out by Fred Donner in his Muhammad and the Believers (2010), which was written for the general public and should be a very easy read. Anyway, Crone's work is still influential, as historians now are far more cautious about traditional sources for Islamic history and are more interested in other non-Arab sources, with the result that some are quite happy to integrate Islam into the world of late antiquity. Sarris' argument above belongs to this strand of thought.

I find this very convincing, and in a previous comment I have argued that based on the events of the sixth century, in which Christianity and Judaism were at the centre of a series of wars within Arabia, Muhammad and his earliest followers cannot but have learned about other faiths and be influenced by their teachings. Even if we follow the most traditional reading of the evidence, Muhammad's wife had a cousin who was a Christian and Medina was home to several clans of Jews. If we take a more revisionist view, then the great powers of late antiquity regularly intervened in Arabia and in the seventh century events in the wider world were still being passed into the Hejaz through merchants, official dispatches, or just ordinary travellers, inevitably shaping what Muhammad taught to his followers. Either way (and I know that I am simplifying things greatly here), Muhammad's teachings constantly engaged with that of Christians and Jews, who were not strangers in Arabia, but members of the same Arabian world, no different from the more familiar pagan Meccan elite or the nomads of the desert.^3

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1. K. van Bladel, 'The Alexander Legend in the Qur’an 18:83–102', in The Qur'an in Its Historical Context (2008).

2. Her posthumous articles present a fine summary of her beliefs; even if they are not always convincing, I think that they will be much discussed in the future: P. Crone, 'Jewish Christianity and The Qurʾān (Part One and Two)', Journal of Near Eastern Studies, 74-75 (2015-2016).

3. But even this is debatable, see G. Hawting, The Idea of Idolatry and the Emergence of Islam: From Polemic to History (1999), which argues that most people were already monotheists at this point.