From 57 to 59 AD, while Paul was imprisoned at Caesarea, Luke was (I believe) traveling around Palestine doing the research for his Gospel's first draft. It's the simplest possible solution, bar none. The only questions are how did Luke compose his text? And from what sources did he work? In answering these questions, the other "Synoptics" may be the solution, instead of the "Problem".
Luke himself knew Mark in Antioch, and Luke relied on Mark's Gospel during his writing process. That much is safe enough. But to go further, Luke had every opportunity [from 57 to 59 AD] to meet Matthew as well. If my theory about Matthew's notes is correct (that Matthew was both the alpha and the omega of the historical composition process for the synoptics) then Luke absolutely would have made sure to procure a copy of those early pages as well. After all, about thirty years after these notes were first made, there naturally would have been many copies. To the early christians, they'd have been famous.
To present hindsight, of course, they'd also have been an unfinished work. In that light, if Matthew's composition was stuck in an early phase for 30 years, and Luke's composition was barely underway during the research phase, then Matthew and Luke would have crossed paths at a time when both men's literary work was ongoing, but not fully developed. At such a time, they would most likely have exchanged source material - oral, written or both. At the very least, we should say Luke copied portions of Matthew's text which Matthew himself would later revise only slightly. (It's also conceivable Matthew picked up fresh research from Luke that Luke wound up discarding.)
Along with Mark and Matthew, who were Luke's other sources? The most likely person to have information about Jesus' early life would be James in Jerusalem. Early-church-politics aside, James should have been at the top of Luke's list for interviewees, and unique details in Luke's Gospel about Nazareth suggests the two men did sit down at some point. Mary, most likely no longer alive, would have been about eighty. Did James report orally, from memory? Perhaps not merely. James may have been writing.
Luke told Theophilus that MANY had attempted to write about Jesus' life and ministry. Luke did not say many succeeded, or finished. Strictly speaking, Luke 1:1 shows that various and multiple writing projects were circulating or underway at that time. Mark's Gospel was finished, but who were the others? As more of an unfinished collection than a cohesive composition, Matthew's notes would be one fitting explanation for Luke's deliberately vague description. Some kind of project by James, yet unfinished, would also make sense. There could even have been some minor 'teachings-only' gospelettes, written by christian pharisees in Jerusalem, probably drawn almost entirely from copies of Matthew's original notes. At any rate, these are possibilities for Luke's literary contemporaries. There may have been more, even if they did not all become sources.
This post wound up majoring on details of my own suggested reconstruction. The next post will be more general, but to the same point.