Chapter III of Redating the New Testament is a bear of a chapter. It's fifty-five pages, making it around 15% of the total monograph. It deals with approximately 50% of the New Testament texts, namely the Pauline epistles. In part due to Robinson's habit of not offering divisions within his chapters, it's not immediately obvious how to divide up discussion of this lengthy chapter. As such, I will offer my own brief rubric for thinking about the chapter, or more specifically the material covered by the material: "core," "prison," and "pastoral" epistles.
The terms "prison" and "pastoral" are not my own, of course. The "prison epistles" refer to Philippians, Ephesians, Colossians, and Philemon: the Pauline epistles that present themselves as written from prison. The term "pastoral epistles" refer by convention to 1 and 2 Timothy and Titus, even though 2 Timothy really doesn't deal with pastoral concerns. The term "core" is my way of referencing those that remain: Romans, 1 and 2 Corinthians, Galatians, 1 and 2 Thessalonians. This particular term is chosen because these texts--especially Romans, 1 and 2 Corinthians, and Galatians--are the texts that receive the greatest attention in Pauline exegesis, and with the exception of 2 Thessalonians are by consensus considered to have been written by Paul himself.
Let us in this post then consider Robinson's treatment of the "core" Pauline epistles. There is little here to which many would object, with perhaps two exceptions. One is that Robinson judges 2 Thessalonians to be an authentic Pauline production, whereas many (but certainly not all) scholars would judge it to be inauthentic. My own feeling is that in 2016 the tide is turning towards authenticity. Douglas Campbell's in many ways revisionist account of Paul's life treats it as authentic, which I think probably reflects a greater openness to Pauline authorship of the disputed epistles even among the more "radical" end of scholarship. More interesting to me is the treatment of Galatians, as it gets at a large number of core chronological concerns.
There has historically been a cleavage between "early" Galatians and "late" Galatians. As a rough definition, "early Galatians" refers to those dating schemes that place Galatians prior to 1 Thessalonians, and "late Galatians" refers to those that place it after. Robinson opts for a "late Galatians" dating, placing Galatians in 56. He is generally impressed by the appearance of shared concerns with Romans, and 1 and (especially) 2 Corinthians. I would acknowledge these, but also press Robinson's own observation that these are not decisive for dating. What strikes me as more decisive is the narrative in Galatians. If we suppose, as Robinson does, that the discussion in Gal. 2:1-10 refers to the Jerusalem council narrated in Acts 15, then we must suppose that 2:11-14 narrates the events that led up to that council. Even Campbell, who programmatically refuses to correlate Pauline and Lukan data for purposes of dating, supposes that vv. 11-14 constitute a "prequel" to 1-10. The difficulty is that I see nothing in Galatians that suggests such a chronological break between 2:10 and 2:11. Indeed, I'm not sure if anyone would think to read Galatians 2 in this fashion, were it not for Acts 15. Gal. 2:11-14 does indeed read like it could be referencing events alluded to in Acts 15:1-2a, and in fact I would argue that it does, but I see absolutely nothing in Galatians 2 to indicate to me that Paul intends us to read these events as prior to the those narrated in vv. 1-10.
If we read Galatians 2:1-10 as referring to events that occurred prior to the events of Acts 15 then we have the salutary benefit not only of more closely adhering the narrative in Galatians, but also of accounting for why Paul only mentions two visits to Jerusalem: just as Acts informs us, as of the time just before the council he had only been to Jerusalem twice since his conversion. We also find that the reason that Paul gives for his second visit is identical with that given for Paul's second visit in Acts: response to a prophecy. Galatians becomes a more coherent narrative, and the Lukan and Pauline data cohere much more fully, if we opt for an "early Galatians," around 47 or 48, written prior to the Jerusalem council of Acts 15.
It should be noted that this is my only serious dispute with Robinson's treatment of the "core" Pauline epistles. It's also one that has relatively little impact beyond Galatians itself. I cannot in fact think of any other text whose date is affected by this difference of opinion. As such, as disagreements go, it's relatively minor.