In order to consider the issues of time travel, it is important do make the distinction between the phenomena (what we perceive) and the noumena (what actually happens in nature, but we don't have access to). In general, time travel implies several logical flaws, which few people seem to take care of, and most conclusions are essentially affected by such flaws.
Things, interactions and causality are facts at a phenomenal level. We perceive them, but that doesn't mean that they are physical facts. Causality is the mental-rational relationship between two events, created only by our understanding. Causality and time generate a chain of events in our understanding, and we create relations between them. For example, a queue of dominoes that fall down in order. If the cause is the Domino-1 fall down, the consequence is the domino-2 fall down, and so on.
When we speak of time travel, at the phenomena level, we are idealizing a change in the causal chain. For example, killing my grandparent in the past is equivalent to the assumption that domino-9 (me) falls down without domino-1 (my grandparent) falling down ever. Is that possible? Not within the rules of causality of this universe, as we've learned them. Remember that the rules of causality govern science (Newton's laws, relativity or Darwin's natural selection are examples).
In consequence, time travel, at the level of our understanding, is just a fallacious idea, and any further conclusion (what happens with causality if I kill my grandparent in the past) is impregnated with such fallacy. In order for time travel to be possible at this level, causality should prove wrong. At this level, time traveling is a nice popular fantasy, equivalent to religious beliefs.
At the noumenal level, causality, things and relationships don't exist as we experience them in our daily lives. Wheeler's experiment proves that the past can be changed. But we don't have the capability to understand the observations of reality at this level. Quantum Mechanics, QM, show that our understanding (that we receive through our senses as phenomenal features) of reality (noumena) is essentially flawed, and the universe behaves in strange ways, which cannot be grasped by our understanding (yes, we have complex formulas, but that doesn't mean that they have a rational meaning). QM is so complex to understand, that we've been forced to assume the possibility of an infinite number of universes (which is not a scientific fact!) to solve the issues of our understanding.
So, it is also a fallacy to conclude that time travel would be possible by amplifying small QM facts, and granting them properties of the macroscopic world (e.g. the whole body of a person would be transported to 1918, with its clothes, memories, the air of his lungs, and the atoms that surround his belly in a radius of 2m).
Then, since time traveling is a fallacious consideration, it is pointless to discuss the consequences of such event.
An equivalent question would be: if Zeus made a miracle in the past, can he undo it without changing my present?