Deliberate mistake: The eyewitness testimony that placed Oswald in the lunchroom at the book despository didn't come to light until the House Select Committee on Assassination in 1978, many years after the Shaw trial.
Factual error: Throughout the movie, it's been said that President John F. Kennedy's motorcade route had been changed at the last minute. This is not true. The route was set in stone for days. In fact, assassin Lee Harvey Oswald supposedly found out that Kennedy was going to ride right past Oswald's work building from a newspaper days before. He rode the same route the newspaper published.
Answer: Lee Harvey Oswald was very sympathetic to the communist ideal, and lived in Russia for a while. He probably adopted the accent as an affectation towards that end.
Grumpy Scot