In Chapter IV of Redating the New Testament, Robinson treats Acts and the Synoptic Gospels. He argues, on the basis of its ending, that Acts was written around c. 62. He argues, as did the late Harnack before him and Colin Hemer after him, that this ending is inexplicable if Paul had already been executed under Nero. I find Robinson, the late Harnack, and Hemer compelling on this matter. I know that there are alternatives (Harnack spent decades trying to defend several of them), but ultimately none seem to deal with the data as neatly as a date of composition c. 62.
When it comes to the Synoptic Gospels, I likewise find his conclusions generally persuasive. He argues that by about c. 40, an early proto-Matthew had emerged in Jerusalem. In the early 40s, in Rome, Mark used this proto-Matthew to produce a proto-Mark, which perhaps went through numerous revisions through to c. 60. Proto-Matthew was also, obviously, a basis for Matthew's Gospel, which also assumes something much like its "final form" c. 60. It was also, to anticipate Chapter X, a source for the Didache. Luke then used these Marks-in-progress and Matthews-in-progress as sources for his own gospel, written in Caesarea during Paul's captivity there. I think that the basic outline best fits the data, although I would make some revisions.
Notably, I am less-than-enthusiastic about his language of "proto-Matthew" and "proto-Mark." No doubt, the Synoptic tradition recurred in numerous forms; the very existence of three Synoptic Gospels demonstrates this fact. I am not entirely persuaded though that any of the gospels went through the numerous "editions" posited in the heydays of source, form, and redaction criticism. This is not to deny that there are variations in the textual traditions for either Matthew's Gospel and Mark's. Of course there are. But that's not quite the same as largely speculative suggestions that portions of the texts that are omnipresent (but not uniform) in the manuscript tradition were missing from a hypothetical "first edition." This skepticism towards proto-gospel hypotheses alters how I would formulate things.
I would agree that the Palestinian church, or I would prefer to say the Jerusalem church, probably generated a body of Jesus material by c. 40. This material would have been the core of what we find in Mark, Matthew, and Luke. My suspicion is that we call "M" and "L," the Special Matthean and Special Lukan traditions, probably were largely present in this material; "M" simply represents that which Matthew drew from this material but Luke did not, and "L" what Luke drew but Matthew did not. Given that we have reason to think that Mark was part of the Jerusalem church during this period (certainly his mother was a significant member of the church by the early 40s), I am favourably disposed to think that he was involved in at least a "clerical" capacity in this initial stage of development. I am inclined to agree with Robinson that Mark and Peter went to Rome in the early 40s, where Peter taught about Jesus. Given Peter's prominent role in the Jerusalem church, his teaching about Jesus can be expected to have largely coincided with that developed in Jerusalem in the 30s. Mark then produced his gospel at the behest of the Roman church. By c. 45 or 46, he and Peter were back in Jerusalem, and Matthew got the idea to improve upon Mark's work by adding material to which Mark, writing in Rome and away from the mother church and concerned to remain close to Peter's versions of events, did not have access or feel free to add. This was perhaps completed by c. 50, as contra Robinson I see Matthew's gospel as more reflective of the Christianity of the late-40s than of the 50s. Subsequently, Luke saw what his fellows had produced, and decided that a comparable text more oriented towards the needs of the Greek mission is in order. Lacking the direct connection with Jesus' followers enjoyed by Mark, secretary to Peter, and Matthew, himself one of the Twelve, Luke decided to go to Palestine and meet with such persons. Thus did he accompany Paul on his final trip to Jerusalem, as indicated by the we-passages in Acts.
Does this narrative run contrary to the consensus view in New Testament studies? Yes. Nonetheless, I think that it is the one that makes the best sense of the various data. It also is one that can accommodate the best insights of modern scholarship. It still allows for the Synoptic tradition to develop over time; it merely places that development at c. 30-60 instead of c. 30-90, and injects greater specificity regarding the identities of those involved in the process. It still allows for such things as social memory to be operative; it merely defines more precisely where, when and among whom such operations were taking place. It still allows for source criticism, but specifies the human relations that facilitated the literary ones. It still recognizes the need to consider Sitze im Leben, but actually names the Sitze in question. Etc. The aim is not to replace such approaches to the gospels, but rather to bring them to perfection through more robust historiography.