Okay...I will assume from the outset that you are not simply re-casting the problems of introducing time-travel to our corner of the universe...
I think that for analytical clarity, it will be necessary to first engage some as of yet unquestioned...issues that underpin your original question.
1. We need to disaggregate different notions of what it is to exist
as something.
meaning a: some
thing (x) exists as a reference to another thing (real or imaginary) (y) to the degree that x shares characteristics with y.
meaning b: (x) exists as (y) to the extent that x points to the same set of phenomena
in the world as does y.*
2. What is a "self", and how does it relate to other entities?
My...overly speculative and...perhaps unnecessarily non-common sensical approach:
my self, empirically, as a discrete "thing", exists as an experience of "now" that is anchored in a moment to moment shift, where each moment births a self-anew. There is an illusion of continuity, born of experiencing memory, born of the physical history of the brain (and the rest of the body).
I will for now remain agnostic on how different selves link (horizontally (other minds simultaneous) and vertically (one mind through time)).
3. What is time, and how does it relate to the universe, in particular conditions of existing, and further in particular to questions of the existence of the self?
And I think these are finally the right questions to allow for speculative answers to the original question.
The experience of time appears to be an illusion built of a perceiving self, engaging the world AND ITSELF at any one moment. However, time also 'exists' 'objectively' (that is, las matter observed as an external entity) insofar as time proves operant in describing causal chains.
SO, to if you changed some of the relevant causes that led to you, as a current self that appears as a particular entity, I would imagine that you would not exist as 'you' do now (in terms of existence meaning a), but this would run along a gradient: to the extent that the stream of certain causes changes, 'you' become different to a certain degree in certain ways.
However, what about the existence of 'you' as referring to a particular 'nexus of consciousness' (meaning b)? This I'm really not sure about at all. . .If we assume the many universes interpretation of QM, and if we assume that each moment within each causal stream in the multiverse corresponds to a particular momentary 'self' that experiences this moment, then disruption of this causal chain would obliterate the particular 'you' of your moment, but that same you would exist elsewhere in the multiverse.**
However, if we wish for there to be some continuous self that maintains integrity through time, I think that 'you' would cease to be, as your particular place in a causal stream moving through time wouldn't exist anymore, as that causal stream would no longer exist.
er...does this make sense at all?
ebola
*I will assume that the divisions between phenomena are fuzzy and that phenomena are nested...hence this "self" is oversimplified when looked at as a unitary thing.
**I think that this works because, per some interpretations of the multiverse, anything that is possible is 'real, actual', but what possibilities will have been realized depend on where in the multiverse you happen to look.