Altering A Timeline

"The Year of the Ransom."

I am not 100% sure of the conclusion of the previous post. How much changes and how much remains the same if a timeline is altered?

Timeline A: the timeline guarded by the Time Patrol;

Timeline B: the timeline that would have resulted if the Exaltationists had stolen Atahualpa's ransom in 1533.

In Timeline B, the Exaltationists either will not exist in the thirty-first millennium or, even if they do exist then, will not travel to 1533 to steal Atahualpa's ransom. However, they will arrive/appear in 1533 and steal the ransom.

The complicating factor is that, in Timeline A, the Exaltationists first establish a base in Machu Picchu in 1610. They intend to steal the ransom in 1533 and then to carry their stolen goods from that year to their base in 1610. Merau Varagan points out that the isolated Machu Picchu will not be noticeably affected by an event, like the disappearance of the ransom, in so recent a past. However, the Exaltationists carrying the stolen ransom futureward from 1533 will travel forward into an altered timeline. It seemed to me that their base, and any of their comrades who had remained at that base, would not exist in the altered timeline. But might they still exist just as the Exaltationists who had stolen the ransom still exist?