This is one of those questions assessed on the balance of probabilities, based on historical context and scraps of information. And what goes for one group might not go for another – each case has to be judged on its own merits on a case by case basis.
One of the points that might be made, to begin with, is that apart from the gospels, there are other kinds of evidence we can look for. One such is archaeological evidence. For instance, in Capernaum in Galilee, no ancient inscriptions has been found by archaeologists from that period. This might well suggest that literacy levels were low. This wasn't the only place frequented by Jesus' group, of course, but it is fair to say that it was an important one to them. So we should be cautious. On the other hand, you would not have had to travel too far to come across the elites, who may well have had slaves, among whom may have been literate slaves (see appendix 2 below), but that is speculation. We need to move to something more relevant to the evidence in the case in hand.
One of the points that I will be making is that Jesus especially, but also his disciples, are portrayed in the gospels (and Acts) as highly motivated people, and this in itself is not average performance, so we are not studying average people per se. This has a bearing on a number of individual points.
Here are a generous ten points which might mean Jesus and/or the wider group of his disciples included some who were above average in their literacy compared to other people of their rural, unprivileged, social background.
- Such recruitment is attested to by Acts 6:7 which claims that “Even a large number of priests became obedient to the faith”.
- A more famous example comes in Acts 15:22-23, which claims literacy by some present in the Jerusalem church in the 40s of the first century, who were literate enough to write down a letter (to be taken to churches where someone had the ability to read the letter): “Then the apostles and elders, with the whole church, decided to choose some of their own men and send them to Antioch with Paul and Barnabas. They chose Judas (called Barsabbas) and Silas, men who were leaders among the believers. With them they sent the following letter…” Content of the letter is then set out in Acts 15.
It is worth mentioning that Jesus will have set out to select apostles who would be capable of going out and preaching his message effectively. So he will definitely have hand-picked people whom he regarded as good communicators. This does not mean they were literate, but it does mean they were probably articulate.
In addition, and this is difficult to measure and therefore unattractive to scholars, some people just have natural talent for communicating that exceeds whatever limited education they may have had. The same thing is remarked, but obviously to different degrees in the case of the peerless writing of Shakespeare (who likely had shortened schooling), the Beatles (they could not read or write music), or indeed the author of John's gospel (likely to have had very limited opportunities as a Galilean fisherman). We can't from this factor make a case for special pleading, but it does caution us to keep reasonably open minds. What will be necessary to develop the argument further, when time allows, would be to look at analogous situations from antiquity where highly motivated people took advantage of opportunities to learn to read.
Sometimes cited as evidence against the literacy of some or all of Jesus' core group of disciples is Acts 4:13. Here, much ink is spilled over whether the Greek word used to describe them - ἀγράμματοί - should be translate as 'illiterate' (Douay-Rheims Bible), 'unschooled' (NIV) or to be interpreted as 'no special training in the Scriptures' (New Living Translation), and there are other suggestions. There is legitimate support for each of these possibilities. But actually, what is easily missed is that Acts could well be deliberately downplaying their degree of literacy as a tactic to imply, "Look, this movement must have its legitimate origin from God because nothing else but help from heaven can explain under-educated people speaking like educated people." This very tactic is used within Islamic apologetics to ascribe divine authority to Mohammed's words by claiming they are the educated-sounding utterances of an uneducated man, and thus surely from heaven. We should be a little bit wary of how keen Luke is to highlight the downplaying of the disciple's education, because Luke's agenda in so doing is to prove that God was on the disciples' side. The impression given by Luke's Acts is that the higher degrees of literacy are in the wider group of early Christians, not the core group. That is not to say that Luke is being untruthful, but that his choice of wording is ambiguous enough to leave us unsure. This is another point that I think is insufficiently addressed by Chris Keith's Jesus Against the Scribal Elite, although there is much in his book that very usefully sets out the data on this subject. So how literate where the disciples? The answer is that it is difficult to say!
To keep this short, I have left out questions such as what we mean by authorship in the case of the gospels. For example, would traditional authorship of, say, the apostle Matthew mean that Matthew wrote the text, or more or less dictated it, or rather supervised some or all of its writing by some literate followers? Each of these options would require higher or lower levels of sophistication of Matthew, but any of these would fit the traditional view. Whereas in the case of Luke's Gospel, where Luke is not one of the core group, it is easier to see him as one of the literate ones in the wider group.
It may be worth adding the the activity of literacy was an enterprise more or less owned by elites, but they didn't necessarily want to do all of the tasks associated with it. in Roman cities, some elites could afford to pay lower status people (even literate slaves) to do the laborious task of copying books by hand, for example. (Freeing the elite to do more interesting things with their time.) So the elites could afford to pay for the upkeep, and perhaps the training, of lower status literate people. (See Loveday Alexander, "Ancient Book Production and the Circulation of the Gospels," in The Gospels for All Christians, ed. Richard Bauckham (Grand Rapids: Eerdmans, 1998), 71-105. E.g. page 88.) What went for Roman cities didn't go for everywhere in the empire, since elites tend to cluster and were not evenly spread geographically, and some cities were more important than others! There is interesting evidence about this at: https://thetextualmechanic.blogspot.com/2020/01/chaerammon-literate-slave.html
The extent to which this scenario would apply to cities in Galilee and Judea is difficult to judge. If we had any evidence of that, or that any within Jesus' circle had been slaves in elite households in such cities, we might be able to explore that further, but we don't have the evidence to go on. We meet Jesus' core followers in the gospels at a stage of life where they are rural fishermen and the like. However, household slaves were not unusual. (This is not like the kind of slavery we saw in the transatlantic slave trade where people were literally stolen. We should think rather of scenarios such as destitute families selling themselves into a different form of servitude.) We might think of Paul's letter to a Christian called Philemon about learning to treat his runaway slave as a brother. Whether the slave, Onesimus, had any such training we are not really in a position to confirm. But my overall point is that it was not impossible by any means for the wider circle to include some who were literate to some or other level.
Appendix 3: What about the literacy of scribes and Pharisees?
On literacy, also see my post: The biblical gospels were all originally anonymous? How historically plausible is that?