All of this works on the assumption that Mark and Luke didn't know each other. The early Christian community was tightly-knit; the lists of personal greetings at the end of various letters should be some clue to that. But the lists of personal greetings also contain evidence that Mark and Luke knew each other directly. Bear in mind that, whether or not you accept the writings attributed to Paul as being from Paul, they still bear record of Mark and Luke being closely associated with each other on several different occasions.
Consider the personal greetings from the letter to Philemon:
Epaphras, my fellow prisoner in Christ Jesus, sends you greetings. And so do Mark, Aristarchus, Demas and Luke, my fellow workers. (Philemon 23-24).The most plain interpretation of that would mean that Mark and Luke knew each other directly. From the language used, they may have known each other as closely as I know my own co-workers.
Consider again the personal greetings from the end of Colossians, where many of the same people also appear in another set of greetings:
My fellow prisoner Aristarchus sends you his greetings, as does Mark, the cousin of Barnabas. ... Our dear friend Luke, the doctor, and Demas send greetings. (Colossians 4:10, 14).Here we have another record of Mark and Luke being together among the companions of Paul at that time.
We have an additional record from Paul's second letter to Timothy:
Only Luke is with me. Get Mark and bring him with you, because he is helpful to me in my ministry. (2 Timothy 4:11)Here we have two passages listing Mark and Luke as being together among the same group in the same city, and a third making arrangements for Mark to come back and join Paul and Luke. Whether or not we accept the attribution of these letters to Paul, they still bear witness to the fact that Luke and Mark were closely associated in the mind of the early church. Based on the early documents we have, it is likely enough that the two knew each other directly. There is no reason to suppose a lengthy delay between Mark finishing his gospel and Luke obtaining a copy.