In the Timeless Universe
In the original timeline, Robert Todd Lincoln rides away from town with General Grant on Apirl 14th, 1865, the day of his father's assassination, and does not witness the event.
In the altered timeline, on April 14th, 1865, Lincoln met Juliet Shakesman, Lucy Preston in disguise, at Ford's Theatre to return tickets for General Grant. He also met John Wilkes Booth, telling him that he owed his family a great debt after Edwin Booth saved his life.
Later, Lincoln and Lucy meet again at the train station. The train had been sabotaged by Garcia Flynn to make sure General Grant attends the play at Ford's Theatre with President Abraham Lincoln. Because of the train malfunction, Robert and Grant decide to attend the play, despite Lucy's recommendation against it. Robert invites Lucy to see the play with them, believing that their two spontaneous meetings in one day must be fate. Lucy accepts the invitation.
While Robert, Grant, Mary Todd Lincoln, and Abraham Lincoln view Our American Cousin at Ford's Theatre, Flynn plots to assassinate both Grant and Abraham Lincoln. Flynn bursts through the door behind Abraham, just as Booth did in the original timeline, but a little bit later. Flynn pistol whips Robert and shoots Abraham in the back of the head. When Flynn tries to kill Grant, he tussles with Robert and Grant until Lucy points a gun at him. Flynn flees after this.
After his father is declared dead, Robert breaks the news to Lucy. Robert thanks her for saving General Grant and leaves, likely to mourn his father's death. 
Notes & References
Robert Todd Lincoln at Wikipedia