Hi Ahanu — interesting question.
I can only speculate, so this is my musing on the subject.
If I were an average Christian in the ancient world, which scrolls would I most likely encounter?
The first encounter would be in the Liturgy. The liturgy would centre on the Eucharist, and comprise of prayers and hymns, readings from the Hebrew Scriptures and the teachings of Jesus — so initially the Gospels.
Regarding what Gospels, then as the book cited, the best evidence we have is surviving elements offers an indication of their relative popularity. Across the second and third centuries, those now considered canonical were the most popular. Fragments of canonical texts outnumber the fragments of apocryphal, heterodox and heresiarch texts by a significant number.
Clement of Alexandria, for example, referred to what we now regard as canonical and apocryphal texts in his writings. If we take a count of citations, his preference is by far for the canon. Clement canonical citations outnumber the apocryphal by about 16 to one. When we look at the Gospels, we find Matthew referenced 757 times, Mark 182, Luke 402 and John 331 times. He cites the apocryphal gospels less than twenty times. (Might I mention Mark with 182 comments, but significantly less than the other gospels.)
Early Church Fathers cite apocryphal writings on very few occasions, compared to their dependence on a proto canon for making their arguments. Notably, The Shepherd of Hermas and The Epistle of Barnabas were held in high regard, but never made the canon. Likewise the letter of Clement of Rome — spot-on as orthodox, but not regarded as canonical.
There was a 'core' canon in place by the middle of the second century.
Additionally, a number of apocryphal writings were expressly condemned. The Gospel of Thomas is never mentioned in any canonical list, not found in any NT manuscript collection, never really figured in theological discussion and often was condemned outright.
Of course some would argue that the paucity of non-canonical texts is because the Church, especially after Constantine, embarked upon a process of their destruction. This really belongs in the realm of a conspiracy theory. Certainly there's no evidence to suggest that apocryphal books were initially regarded as Scripture and later were weeded out and destroyed. The more likely story is that while some books, like those mentioned above, were given high status, the evidence suggests that the books we now regard as canonical were preferred from the start and indeed that's why they became the canon. While some might suggest the canon was some arbitrary creation of a later era, the evidence suggests the later church simply affirmed what had been the case for centuries.
No doubt there was a demand for books, and people ready to meet that demand. So the appearance of 'fake gospels' is not unlikely. That's why the early Church fathers were quite demanding in what passed as authentic and what was fake. Undoubtedly the average Christian reader might not be so discerning. There was a great demand for infancy narratives, for example. But these texts would have gone into private collections rather than official libraries, and would have been secretly preserved and therefore we should expect to find remains in proportion to their popularity.
Would a Christian in Alexandria encounter a different collection of Christian texts from one in Rome?
I'd say s/he'd encounter more, as Alexandria was one of the centres of thought generally, and Christian thought especially (along with Antioch), so there would have been a demand there for books, and traders answering that demand. Rome was never really a theological centre, more the pastoral centre.
Let's assume I am illiterate since I am your average Christian, so I listen to them instead of read them.
In which case it would be the texts read in the Liturgy and the supported teachings. The Arian dispute, for example, was triggered by complaints from your 'average illiterate Christian' that what their presbyter Arius was teaching was not the same as what they had learned in their catechetical education.