Can we use Heisenberg Uncertainty Principle to prove that time travel to past is impossible?

Contrary to popular belief, the HUP is not a principle about the accuracy of a measurement. The HUP is simply a statement that relates the spread of position measurements to the spread of momentum measurements of similarly prepared systems. It is a statistical principle about multiple measurements and their standard deviations; it is not a principle that applies to single measurements of one system.

Furthermore, this idea that the HUP means that "the particle has an exact position and momentum, but we just don't know what they are" is not correct. The uncertainties discussed in the HUP arise purely from the postulates of QM and have nothing to do with what we know about the particle or how accurately we measure its position or momentum. Many QM interpretations would even say that before measurement the particle doesn't even have a defined position or momentum at all.

Therefore, your premise is flawed purely from a misunderstanding of what the HUP actually says and how it applies to quantum systems.

Of course, it is an interesting thing to think about what would be the outcome of a quantum measurement if you were to travel back in time and repeat the measurement again in precisely the same manner. Unfortunately, I don't think anything like this can be experimentally tested (at least for now ;) ), so anything about this point would be pure speculation.

The key point about the Hiesenberg uncertainty principle is that the measurement, itself, changes the quantum state of the system. So, if you went and collapsed the system onto a momentum state at time $t_0$, and then went back to some time $t_{-1} < t_0$ and tried to collapse the same system onto a position state, you would decohere the momentum of the particle, and the second experiment to measure the momentum of the particle would give a different answer.

No, the uncertainty principle is purely a property of the state of a system. Given two observables $A$ and $B$, you have a statement of the form

$$ Delta A Delta Bge k$$

for some constant $k$. This tells you that if you are able to guess the outcome of an $A$ measurement with good accuracy ($Delta A$ is small), then you wouldn't be able to guess the outcome of a $B$ measurement accurately ($Delta B$ is large). If you measure $A$ and then you "go back in time" (by which I mean you revert the system to its original state $psi$), you have no more information of what the outcome of either measurement will be, and the same uncertainty principle will apply. A new measurement of $A$ would yield a new value uncorrelated with the one you previously got, and nothing would change.

In other words, the question is: if I have a system in the state $psi$ and I measure some observable $A$, then the state of the system becomes an eigenstate of $A$. I now apply some transformation to the system such that its state gets reverted back to $psi$. Does my ability to guess the outcome of a measurement of an observable $B$ change? Clearly not.

Please don't accept this answer, I just want to include it here for the next reader/asker of this question.

So @user2723984's answer gets to the heart of the matter mathematically and @biophysicist's answer gives you some intuitive understanding of what that means.

After reading the OP's comments it seem you still wanted to try to recover the thought experiment and make it work using time travel to create some kind of paradox.

The best I could find was we have a system $psi$ we are observing and at $T$ we highly constraint the standard deviation of the momentum of the wave function $psi$, then some time passes, and we go back in time and highly constrain the position instead, and declare "oh but at time T both the position and momentum were known!" but that's not very interesting since though the time is shared the timelines themselves are different. It looks to me (though I am not confident if this is true) whether or not you time travel, at each moment of time locally speaking the heisenberg uncertainty principle holds true. I.E. it will take more than just time travel to break it locally.

If someone thinks they can violate the heisenberg uncertainty principle assuming access to time travel i'd be curious to see it in the comments.

That is a beautiful idea. And I believe it is true. If time travel were possible, you could measure (for each) particle observable A and remember which ones lead to what result. Then travel back and do the same for observable B. Then travel back again and you remove the particles from the ensemble based on your knowledge. This will allow you to create an ensemble which violates H.U.